LE  MEN  TART 
FUNCTIONS 


UC-NRLh 


IN  MEMORIAM 
FLORIAN  CAJORl 


ELEMENTARY     FUNCTIONS 

AND 

APPLICATIONS 


BY 

ARTHUR   SULLIVAN   GALE,   Ph.D. 


AND 


CHARLES   WILLIAM   WATKEYS,   A.M. 

PROFESSORS   OF   MATHEMATICS 
IN   THE    UNIVERSITY   OF   ROCHESTER 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 


43 


Copyright,  1920 

BY 

Henry  Holt  and  Company 


•    < «  •    • 


CAJORl 


r 


PREFACE 

This  book  presents  a  coherent  year's  work  in  mathematics 
for  college  freshmen,  consisting  of  a  study  of  the  elementary 
functions,  algebraic  and  transcendental,  and  their  applications 
to  problems  arising  in  various  fields  of  knowledge.  The 
treatment  is  confined  to  functions  of  one  variable,  with  inci- 
dental exceptions,  and  complex  values  of  the  independent  and 
dependent  variables  are  excluded.  The  subject  matter  includes 
the  essentials  of  plane  trigonometry  and  topics  from  advanced 
algebra,  analytic  geometry,  and  calculus. 

The  text  is  the  result  of  experiments  beginning  in  1907-8. 
It  has  been  used  in  the  classroom  since  1913-14,  and  each  year 
extensive  revisions  have  been  made.  Hence  the  content  of 
the  course,  the  order  of  topics,  and  the  manner  of  presentation 
are  based  upon  the  experience  of  several  years. 

The  unity  of  the  course  is  gained  by  an  explicit  analysis  of 
the  functions  studied,  which  enables  the  student  to  compre- 
hend the  purpose  of  the  course  as  a  whole  and  the  nature  of 
the  investigation  of  properties  of  functions  of  a  given  type. 
This  analysis  consists  of  three  parts: 

First.  Relations  between  a  given  function  and  its  graph 
(see  pages  42  and  274).  Most  of  these  relations  are  considered 
in  the  first  chapter  so  that  at  the  start  the  student  is  made 
aware  of  a  number  of  questions  which  will  be  investigated  in 
studying  a  particular  type  of  functions. 

Second.  Relations  between  pairs  of  functions  and  their 
graphs  (see  page  152).  These  geometric  transformations  are 
introduced  in  connection  with  simple  algebraic  functions  so 
that  they  are  familiar  tools  by  the  time  they  are  needed  for  the 
study  of  transcendental  functions. 

918206 


iv  PREFACE 

Third.  Analogous  properties  of  functions  which  have  no 
immediate  graphical  interpretation.  Several  properties  of  x^ 
are  grouoed  together  on  page  153  in  order  to  indicate  further 
questions  which  should  be  investiagted  in  studying  trans- 
cendental functions. 

Emphasis  is  also  placed  on  characteristic  properties  which 
distinguish  one  class  of  functions  from  another. 

A  very  large  group  of  freshmen  taking  mathematics  do  not 
continue  the  study  of  this  subject  in  the  following  years,  and 
the  needs  of  these  students  have  received  primary  considera- 
tion. For  the  general  student,  the  interpretation  of  a  graph, 
the  fundamental  concepts  of  the  calculus,  and  the  usefulness 
of  mathematics  are  of  fundamental  importance.  Fortunately 
these  matters  are  also  important  for  the  student  of  mathe- 
matics, and  experience  with  the  text  has  shown  that  it  is  pos- 
sible in  the  second  year  satisfactorily  to  complete  the  usual 
first  courses  in  analytic  geometry  and  calculus. 
!,'  To  show  the  usefulness  of  mathematics,  a  wide  range  of 
problems  which  deal  with  matters  of  interest  to  the  student 
have  been  introduced,  although  exercises  which  require  con- 
siderable instruction  in  other  subjects  have  been  avoided. 
AppHcations  to  the  solution  of  problems  in  mechanics,  physics, 
chemistry,  economics,  and  other  subjects  have  been  scattered 
throughout  the  course.  The  analysis  of  a  problem  in  a  field 
other  than  mathematics  is  usually  more  difficult  for  a  freshman 
than  the  solution  after  the  conditions  and  requirements  have 
been  stated  in  mathematical  language.  But  from  the  broad 
standpoint  in  which  mathematics  appears  as  part  of  an  educa- 
tional system,  the  training  in  such  analysis  is  as  important  as 
the  development  of  the  mathematical  processes  to  which  the 
analysis  leads.  The  obligation  resting  upon  the  teacher  of 
mathematics  to  develop  this  power  of  analysis  is  increased 
by  the  proneness  of  other  teachers  to  tread  very  lightly  on  the 
mathematical  aspects  of  their  own  subjects,  and  it  is  quite 
possible  that  this  inclination  on  the  part  of  others  is  partly 
due  to  the  failure  of  mathematicians  to  emphasize  the  appU- 
cations   sufficiently.     Simple   appUed   problems   may  furnish 


PREFACE  V 

drill  in  mathematical  technique,  with  added  interest,  and  with 
but  slightly  increased  difficulty.  Simultaneously,  they  afford 
some  training  in  analysis. 

Attention  is  called  to  the  following  features: 

1.  The  chapter  on  the  theory  of  measurements  gives  an 
outline  of  statistical  methods  which  are  used  in  many  fields 
such  as  economics,  biology,  physics,  education,  etc.  It  is, 
perhaps,  not  too  much  to  say  that  the  average  college  graduate 
will  find  more  use  for  this  topic  than  for  any  included  in  the 
traditional  freshman  course.  The  treatment  given  here  is 
intermediate  between  the  books  on  statistics  which  presuppose 
very  little  mathematical  theory  and  those  which  the  mathe- 
matical prerequisites  render  unsuitable  for  the  average  college 
man. 

2.  Emphasis  is  placed  upon  the  determination  of  a  function 
from  a  given  table  of  empirical  data.  In  problems  of  this  sort, 
which  illustrate  an  important  method  of  discovery  in  science, 
there  is  an  element  of  general  culture  which  is  too  often  neg- 
lected in  elementary  courses  in  the  various  sciences  on  account 
of  the  mathematics  involved. 

3.  Graphical  methods  of  analyzing  problems  are  used  freely. 

4.  Graphical  methods  are  used  in  problems  which  can  be 
solved  by  straight  Unes,  and  an  algebraic  solution  is  then 
obtained  by  finding  the  equation  of  certain  Unes  used  in  the 
graphical  solution. 

5.  Use  is  made  of  the  graphs  of  the  trigonometric  functions 
in  tying  together  and  affording  the  means  of  recalling  many 
properties  of  these  functions.  This  use  of  the  graphs  is  merely 
a  part  of  the  general  point  of  view  of  the  course. 

6.  Trigonometric  analysis,  the  most  abstract  topic  included, 
is  postponed  until  late  in  the  course. 

7.  The  introduction  of  a  considerable  amount  of  the  ele- 
mentary portions  of  the  calculus  gives  the  general  student  a 
knowledge  of  the  importance  and  utility  of  the  fundamental 
ideas  of  derivative  and  integral. 

8.  The  average  rate  of  change  of  a  function  is  introduced 
at  the  start,  and  it  is  used  in  studying  the  linear  function. 


vi  PREFACE 

The  rate  of  change  is  introduced  informally  in  connection 
with  the  quadratic  function,  while  formal  treatment  is  post- 
poned until  later.  The  difficulties  in  grasping  the  concept  of 
a  derivative  are  thus  separated,  and  time  for  thorough  assimi- 
lation is  afforded. 

9.   Integration  is  used  to  obtain  the  volumes  of  a  pyramid, 
cone,  and  sphere. 

10.  Tables  of  squares,  square  roots  etc.,  that  is,  tables  of 
functions  famiHar  to  the  student,  are  used  in  advance  of  othisr 
tables.  The  table  of  logarithms  is  introduced  as  a  general 
tool,  and  it  is  not  regarded  as  something  to  be  used  primarily 
with  the  trigonometric  functions.  Use  is  made  of  various 
tables  throughout  a  large  part  of  the  course,  so  that  the  student 
acquires  faciUty  in  their  use  and  in  the  selection  of  the  most 
suitable  table  for  a  given  computation. 

11.  The  use  of  the  slide  rule,  and  of  logarithmic  and  semi- 
logarithmic  cross  section  paper,  is  explained  in  connection  with 
the  logarithmic  function. 

12.  The  functions  a;"  and  6^,  which  occur  frequently  in  the 
appHcations  of  mathematics,  are  treated  at  some  length. 

13.  An  effort  has  been  made  to  render  expHcit  the  purpose 
of  the  various  parts  of  the  course. 

14.  Formal  proofs  of  a  number  of  theorems  are  omitted, 
and  some  are  assumed  without  proof.  The  appeal  to  the 
intuition  imderlying  most  of  these  assumptions  is  justified  by 
the  behef  that  the  logical  presentation  of  these  theorems  re- 
, quires  a  foundation  too  abstract  for  the  general  student,  or 
too  cumbersome  for  the  purpose  to  be  served. 

15.  The  course  includes  more  than  a  year's  work  so  that 
teachers  have  an  opportunity  for  choice  of  topics,  and  abundant 
naaterial  is  provided  for  selected  sections  which  progress  more 
rapidly  than  the  average. 

16.  The  course  increases  in  difficulty  with  a  corresponding 
increase  in  interest  and  gain  in  power  on  the  part  of  the  student. 

17.  A  very  wide  range  of  problems,  varying  in  difficulty, 
makes  it  possible  for  the  instructor  to  emphasize  different 
aspects  of  the  subject,  to  select  exercises  suitable  for  students 


PREFACE  vu 

of  different  abilities,  and  to  assign  different  sets  of  exercises  in 
different  years. 

18.  Many  of  the  exercises  in  the  later  chapters,  while  dealing 
with  the  newer  topics,  are  constructed  to  afford  review  of 
principles  presented  earUer  in  the  course,  and  to  correlate 
various  parts  of  the  subject. 

We  thank  several  friends  for  the  inspiration  of  their  interest 
and  for  their  suggestions.  We  also  thank  the  Trustees  of  The 
University  of  Rochester  for  making  it  possible  for  us  to  use 
the  text  in  the  classroom  throughout  our  experimentation,  in 
which  we  have  participated  equally.  Combined  courses  are 
still  to  be  regarded  as  in  the  experimental  stage,  but  it  is  our 
conviction  that  they  are  fundamentally  sound,  and  we  shall 
feel  well  repaid  if  this  volume  contributes  something  of  value 
to  their  development. 

Arthur  Sullivan  Gale 
Charles  William  Watkeys 
The  IlNrvTERsiTY  of  Rochester 
May,  1920. 


CONTENTS 

PAai5 

Some  Principles  of  Algebra  and  Geometry xv 

CHAPTER  I 

SECTION  FUNCTIONS,   EQUATIONS,   AND   GRAPHS 

1.  Comparison  of  the  Reasoning  in  Natural  Science  and  in 

Mathematics 1 

2.  Example  of  the  Utihty  of  Mathematics  in  Science 2 

3.  Forms  in  which  Data  are  Recorded 3 

4.  Variable.     Function 4 

5.  Notation  for  a  Function 8 

6.  Determination  of  the  Function  which  Expresses  the  Func- 

tional Relation  between  Two  Variables    8 

7.  Graphical  Representation.     Directed  Lines 12 

8.  Rectangular  Coordinates 14 

9.  Graph  of  a  Function 16 

10.  Discussion  of  the  Table  of  Values 21 

11.  Functions  becoming  Infinite.     Asymptotes 25 

12.  Variation  of  a  Function 28 

13.  Average  Rate  of  Change  of  a  Function 34 

14.  Classification  of  Functions 38 

15.  Summary 41 

CHAPTER  II 

LINEAR   FUNCTIONS 

16.  Uniform  Rate  of  Change 46 

17.  Characteristic  Property  of  a  Straight  Line 49 

18.  Slope  of  a  Straight  Line 51 

19.  Graphical    Solution    of     Problems     Involving     Functions 

which  Change  Uniformly 54 

ix 


X  CONTENTS 

20.  Graph  of  the  Linear  Function  mx  -{- b ...  56 

21.  Variation 61 

22.  Uniform  Acceleration 63 

23.  Equation  of  a  Straight  Line 66 

24.  Application  to  the  Solution  of  Problems 69 

25.  Remarks  on  Measurements 72 

26.  Possible  Errors  in  Arithmetic  Calculations.     Abridged  Mul- 

tipHcation  and  Division 74 

27.  Empirical  Data  Problems , .  78 

CHAPTER  III 

ALGEBRAIC   FUNCTIONS 

28.  Introduction 87 

29.  Graph  of  x^ 87 

30.  Graphs  of  ax^  and  af(x) 88 

31.  Translation  of  the  Coordinate  Axes 89 

32.  Instantaneous  Velocity 93 

33.  Rate  of  Change.     Slope  of  Tangent  Line 94 

34.  Graph  of  the  Quadratic  Function  ax^  +  bx  +  c 98 

35.  Empirical  Data  Problems 104 

36.  The  Function  x" 107 

37.  Tables  of  Squares,  Cubes,  Square  Roots,  Cube  Roots,  and 

Reciprocals 107 

38.  Graph  of  x",  w  >  1 110 

39.  Graph  of  x",  0  <  w  <  1.    Graphs  of  Inverse  Functions  . .  113 

40.  Graph  of  x",  n  <  0.    Graphs  of  Reciprocal  Functions 117 

41.  Summary  of  Graph  of  x" 119 

42.  Interpolation 121 

43.  Variation 125 

44.  Empirical  Data  Problems 127 

ax  +  b 

45.  The  Linear  Fractional  Function  — — : 131 

ex  +  a 

46.  Integral  Rational  Functions 133 

47.  The  Remainder  Theorem 133 

48.  Synthetic  Division 134 

49.  Graph  of  a  Polynomial 136 

50.  Extent  of  the  Tables 137 

51.  Solution  of  Equations.     Rational  Roots 140 

62.  Translation  of  the  ^/-axis 144 


CONTENTS  xi 

53.  Horner's  Method  of  Solution  of  Equations 147 

54.  Graph  of  the  Function  /  (ax) 150 

55.  Related  Functions  and  their  Graphs 151 

56.  Some  Operations  of  Algebra  regarded  as  Properties  of  Func- 

tions    153 


CHAPTER  IV 

TEIGONOMETRIC   FUNCTIONS 

57.  Introduction 158 

58.  Angles  of  any  Magnitude 164 

59.  Trigonometric  Functions  of  any  Angle 165 

60.  Radians 170 

61.  Graphs  of  the  Trigonometric  Functions 172 

62.  Functions  of  Complementary  Angles 177 

63.  Tables  of  Trigonometric  Functions 178 

64.  Solution  of  Right  Triangles 180 

65.  AppHcations 183 

66.  Parallelogram  Law  —  Velocities,  Accelerations,  Forces 185 

67.  Conditions  of  Equilibrium  of  a  Particle 187 

68.  Functions  of  n90°  ^d 192 

69.  Apphcation  to  the  Use  of  Tables 196 

70.  IncUnation  and  Slope  of  a  Straight  Line 199 

71.  Law  of  Sines 201 

72.  Law  of  Cosines 202 

73.  Solution  of  Oblique  Triangles 203 

74.  Inverse  Trigonometric  Functions 209 

CHAPTER  V 

EXPONENTIAL  AND   LOGARITHMIC   FUNCTIONS 

75.  Introduction 214 

76.  Graph  of  the  Exponential  Function  6*,  6  >  1 216 

77.  Properties  of  the  Exponential  Function  6*,  6  >  1 217 

78.  Computation  by  Means  of  an  Exponential  Function 219 

79.  The  Logarithmic  Function,  the  Inverse  of  the  Exponential 

Function 221 

80.  Graph  of  the  Logarithmic  Function 221 

81.  Properties  of  the  Logarithmic  Function,  log6  x,b  >  1 222 


xii  CONTENTS 

82.  Common  Logarithms 225 

83.  Computation  by  Means  of  Common  Logarithms 229 

84.  Solution  of  Triangles 233 

85.  Exponential  Equations 237 

86.  Compound  Interest 241 

87.  Annuities 244 

88.  Graph  of  the  exponential  function  kb"' 248 

89.  The  Logarithmic  Scale 249 

90.  Empirical  Data  Problems 256 

CHAPTER  VI 

DIFFERENTIATION   OF  ALGEBRAIC  FUNCTIONS 

9L  Introduction 264 

92.  Limits 265 

93.  Derivative  of  a  Function 267 

94.  Fundamental  Formulas  for  Differentiation 269 

95.  Derivative  of  a  Polynomial 271 

96.  Corresponding  Properties  of  a  Function,  its  Graph  and  its 

Derivative 272 

97.  Velocity  and  Acceleration 276 

98.  Derivative  of  a  Rational  Function 278 

99.  Derivative  of  an  Irrational  Function 279 

100.  Equations  of  Tangent  and  Normal  Lines 281 

101.  Problems  in  Maxima  and  Minima 285 

102.  Related  Rates 289 

103.  Small  Errors 291 

104.  Approximate  value  o{f{x-\-  Ax) 296 

CHAPTER  VII 

INTEGRATION 

105.  Introduction 301 

106.  Area  under  a  Curve 304 

107.  Motion  in  a  Straight  Line 308 

108.  Motion  in  a  Plane , 311 

109.  Volume  of  a  Right  Prism 315 

110.  Volume  of  a  Right  Circular  Cylinder 318 

111.  Volume  of  a  Pyramid 320 

112.  Volume  of  a  Solid  of  Revolution 323 


CONTENTS  xiii 

113.  Volume  of  a  Cone  of  Revolution 325 

1 14.  Volume  of  a  Sphere 327 

115.  Area  of  a  Sphere 328 


CHAPTER  VIII 

PROPERTIES   OF  TRIGONOMETRIC   FUNCTIONS 

Logarithmic  Solution  of  Triangles,  Cases  III  and  IV 

116.  Introduction 332 

117.  Fundamental  Trigonometric  Relations 332 

118.  Trigonometric  Equations 334 

119.  Trigonometric  Identities 336 

120.  Functions  of  the  Sum  of  Two  Angles 338 

121.  Functions  of  the  Difference  of  Two  Angles 341 

122.  Functions  of  Twice  an  Angle,  or  the  Functions  of  Any 

Angle  in  Terms  of  Half  the  Angle 342 

123.  Functions  of  Half  an  Angle,  or  Functions  of  Any  Angle  in 

Terms  of  Functions  of  Twice  the  Angle 343 

124.  Sum  and  Difference  of  the  Sines  or  Cosines  of  Two  Angles  345 

125.  Logarithmic  Solution  of  Triangles,  Case  III 346 

126.  Logarithmic  Solution  of  Triangles,  Case  IV 348 

127.  Miscellaneous  Identities  and  Equations 351 

128.  Differentiation  of  Trigonometric  Functions 352 

129.  Graph  of  the  Function  a  sin  (6x  +  c).    Harmonic  Curves .  .  358 

130.  Empirical  Data  Problems 361 

CHAPTER  IX 

THEORY   OF  MEASUREMENT 

131.  Statistical  Methods 365 

132.  Permutations 366 

133.  Combinations 367 

134.  The  Binomial  Expansion 370 

135.  Probabihty 373 

136.  Compound  Events ' 375 

137.  Mortahty  Tables 379 

138.  Frequency  Distributions 382 

139.  Averages 388 


xiv  CONTENTS 

140.  Measures  of  Variability 398 

141.  Equation  of  the  Frequency  Curve  Representing  a  Sym- 

metrical Distribution 404 

142.  The  Probable  Error 411 

143.  Least  Squares 416 

144.  Correlation 420 


SOME    PRINCIPLES    OF    ALGEBRA   AND    GEOMETRY 

1.  Classification  of  the  Numbers  of  Algebra. 

/  Integral  (1,  2,  -3,  etc.) 
[  I^ational  ^  Fractional  (f,  -h,  etc.) 

Real  1  _      _ 

[  Irrational  (V2,  V3,  x,  etc.) 

Complex  (or  imaginary) ,  which  will  not  be  considered  in  this  course. 

2.  Laws  of  Addition  and  Multiplication. 

(a)  Commutative  laws:  a  +  b  =  b  +  a.     ah  =  ha. 

(b)  Associative  laws:  a+  (b  +  c)  =  (a  +  h)  +c.    a{hc)  =  {ab)c. 

(c)  Distributive  law:  a(b  +  c)  =  ah  +  ac. 

3.  Zero. 

(a)  0X0  =  0. 

(b)  If  o  X  6  =  0,  then  a  =  0  or  6  =  0. 

(c)  It  Is  impossible  to  divide  by  zero,  for  the  quotient  of  a  by  zero, 
if  it  existed,  would  be  a  number  q  such  that  g  x  0  =  a.  But  as  g  x  0  =  0, 
by  (a),  we  have  a  contradiction,  and  hence  division  by  zero  must  be 
excluded. 

4.  Fractions. 

(a)  The  value  of  a  fraction  is  unchanged  if  numerator  and  denomina- 
tor are  divided  by  the  same  number  not  zero.  This  enables  us  to  "cancel " 
a  common  factor  of  the  numerator  and  denominator. 

(b)  The  value  of  a  fraction  is  unchanged  if  numerate"  and  denominator 

are  multiplied  by  the  same  number,  that  is,  r  =  — r  • 

This  gives  the  rule:  To  simplify  a  given  complex  fraction,  multiply 
numerator  and  denominator  by  the  least  common  denominator  of  the 
fractions  occurring  in  the  numerator  and  denominator  of  the  given  frac- 
tion.   Thus  if  we  multiply  numerator  and  denominator  of  the  complex 

e      .-      a      b  .       ,  .     .  bmx  +  amy 

fraction  -r-  by  abm  we  get  at  once   -r o* 

d      ''  ^  abcm  +  ahd 

c-\ 

m 


(c)  Addition  and  subtraction:    r+  7=  — Tj — * 

b     d  bd 


rvi    SOME  PRINCIPLES  OF  ALGEBRA  AND  GEOMETRY 

(d)  Multiplication:   r  X  7  =  r^- 

o       a      bd 

/  \  T\'  '  •         CL      c     ad 

(e)  Division:   r  ^  -7  =  . —  • 

0      a      be 

(f)  The  reciprocal  of  a  given  number  is  a  second  number  such  that  the 
product  of  the  two  is  unity.     Thus  the  reciprocal  of  a  is  -• 

6.  Rules  for  Signs. 
(a)  (-  a)  (-  b)  =  ab.       (b)  (-  o)  (6)  =  -  ab. 
,  .    —a      a  ...  —a       a         a  ,  .  /,        x 

(^)  Zb^b'  ^"^^  T^  =6"~6*  (e)  a-(b-c)^a-b  +  c. 

6.  Factors. 

(a)  a2  -  62  =  (a  +  b)  (a  -  6). 

(b)  o3  +  63  =  (a  +  6)  (a2  -  a6  +  62). 

(c)  o'  -  6»  =  (a  -  6)  (a2  +  a6  +  62). 

(d)  a«  +  6«  =  (a  +  6)  (a^-i  -  a''-26  +  a«-'62  -  .  .  .  +  6«-0,  n  odd. 

(e)  a"  -  6"  =  (a  -  6)  (a^-^  +  a"-26  +  a^'-^b-  +  .  .  .  +  6»-i). 

(f)  ofi+  (a  +  b)x  +  ab  =  (x  +  a)  {x  +  6). 

7.  Binomial  Theorem. 

(a  +  6)-  -  a-  +  na"-.6  +  ^^^  a»-  6^  +  "^""/''""^^   a-  W 
+  .--  +  6'». 
The  coefficient  of  the  (r  +  l)st  term  is  >^  (^  -  1)(^  "  2)  •••  (n  -  r  +  1) 

Special  cases:  n  =  2,  (a  +  6)2  •=  a^  +  2ab  +  ¥. 

n  =  3,  (a  +  6)3  =  a'  +  3a26  +  3a62  +  b\ 

8.  Powers  and  Roots. 


(a)            aO  =  l. 

(b)     «-"  =  ^- 

(c)   (a*")"  =  a»'"» 

-      «/- 
(d)           a«  =  VaP. 

(e)  a'"-a''  =  a"'+«. 

(f)    ^  =  a'«-». 

(g)  Va  VF  =  Vab. 

w  5=./? 

(i)  To  rationalize  the  denominator  (or  numerator)  of  a  fraction  con- 
taining a  square  root,  multiply  numerator  and  denominator  by  the  quantity 
obtained  by  changing  the  sign  of  the  radical  in  the  denominator  (or  nu- 
merator). 

9.   Equations. 

(a)  If  equals  be  added  to,  or  subtracted  from,  equals  the  results  are 
equal. 

(b)  If  equals  be  multiplied  or  divided  by  equals,  the  results  are  equal. 


ELEMENTARY  FUNCTIONS  xvii 

Do  not  divide  both  sides  of  an  equation  by  a  quantity  until  the  pos- 
sibihty  of  division  by  zero  has  been  excluded. 

(c)  Like  powers,  or  roots,  of  equals  are  equal. 

(d)  To  simplify  an  equation  containing  fractions,  multiply  both  sides 
by  the  lea^t  common  denominator  of  the  fractions. 

(e)  To  simpHfy  an  equation  containing  square  roots,  transpose  to  one 
side  all  the  terms  except  a  single  radical,  and  square  both  sides. 

(f)  To  solve  the  quadratic  equation  ax^  +  bx  +  c  =^  0,  factor  the  left- 
hand  member,  set  each  factor  equal  to  zero,  and  solve  for  x. 

If  the  leflr-hand  member  cannot  be  easily  factored,  transform  the  equa- 
tion to  the  form 


and  complete  the  square  by  adding  to  both  sides  the  square  of  half  the 
coefficient  of  x.     Then  extract  the  square  root  of  both  sides  of  the  result- 
ing equation,  and  solve  for  r. 
(g)  The  roots  of  the  equation 


,     ,               »                             -b  ^  V62  -  4ac 
ax^  +  bx  -\-c  =  0        are        x  =  ^ 

The  roots  will  be  real  and  imequal,  real  and  equal,  or  imaginary  accord- 
ing as  the  discriminant  b^  -  4ac  is  positive,  zero,  or  negative. 

(h)  Simultaneous  equations  in  two  variables  x  and  y. 

If  both  equations  are  linear,  to  eliminate  y,  multiply  each  equation  by 
the  coefficient  of  y  in  the  other,  and  subtract  the  results. 

If  one  equation  is  linear  and  the  other  quadratic,  to  eliminate  one  vari- 
able, solve  the  linear  equation  for  x  or  y  (whichever  is  easier)  and  sub- 
stitute in  the  quadratic  equation. 

10.  Arithmetical  Progression. 
In  an  arithmetical  progression, 

a,  a  +  d,  a  +  2dj  a  +  3d,  .  .  .  J 

each  term  is  obtamed  from  the  preceding  by  adding  a  constant  quantity 
The  nth  term  is  I  =  a  +  (n  -  l)d. 

It 
The  sum  of  n  terms  is  S  =  j^  {a  +  I). 

The  arithmetic  mean  between  a  and  bis  A  =  I  (a  +  b). 

11.  Geometric  Progression. 
In  a  geometric  progression, 

a.  ar,  ar^,  ar^,  .  .  .  , 

each  term  is  obtained  from  the  preceding  by  multiplying  by  a  constant 
quantity. 


xviii    SOME  PRINCIPLES  OF  ALGEBRA  AND  GEOMETRY 

The  nth  term  is  I  =  ar"^'^. 

The  sum  of  n  terms  is  <S  =  7  • 

r  -  1 

The  geometric  mean  between  a  and  6  is    G  =  V06. 
If  the  numerical  value  of  r  is  less  than  unity,  as  n  increases  indefinitely 
the  sum  of  n  terms  approaches  the  limit  -z 

12.  Parallel  Lines. 

Two  lines  are  parallel  if  the  alternate-interior  angles  are  equal,  or  if 
the  exterior-interior  angles  are  equal,  and  conversely. 

13.  Triangles. 

(a)  The  bisector  of  the  vertex  angle  of  an  isosceles  triangle  bisects  the 
base  perpendicularly. 

(b)  In  any  plane  triangle  the  sum  of  the  anglfes  is  180°. 

(c)  The  exterior  angle  of  a  triangle  is  equal  to  the  sum  of  the  two  op- 
posite interior  angles. 

(d)  In  a  right  triangle  the  square  on  the  hypotenuse  is  equal  to  the  sum 
of  the  squares  on  the  other  two  sides. 

14.  Similar  Polygons. 

(a)  Similar  polygons  are  polygons  which  have  their  angles  respectively 
equal  and  their  homologous  sides  proportional. 

(b)  If  two  triangles  have  the  angles  of  the  one  equal  respectively  to 
the  angles  of  the  other,  the  triangles  are  similar. 

(c)  If  two  triangles  have  an  angle  of  one  equal  to  an  angle  of  the  other, 
and  the  including  sides  proportional,  the  triangles  are  similar. 

(d)  If  two  triangles  have  their  sides  respectively  proportional  they  are 
similar. 

(e)  The  areas  of  two  similar  polygons  are  to  each  other  as  the  squares 
of  any  two  homologous  sides. 

15.  Circles. 

(a)  An  angle  at  the  center  of  a  circle  is  measured  by  the  intercepted  arc. 

(b)  An  inscribed  angle  of  a  circle  is  measured  by  one  half  the  intercepted 
arc. 

(c)  The  circumference  of  a  circle  is    C  =  27rr. 

(d)  The  area  of  a  circle  is  A  =  irr^. 

22 

(e)  Approximate  values  of  tt  are  y  and  3.1416. 

(f )  A  sector  of  a  circle  is  a  figure  formed  by  two  radii  and  their  inter- 
cepted arc. 

The  area  of  a  sector  is  A  =  ^  arc-r. 

(g)  A  segment  of  a  circle  is  the  figure  formed  by  a  chord  and  its  inter- 
cepted arc. 


ELEMENTARY  FUNCTIONS 


XIX 


16.   Operations  on  Lines. 

(a)  Addition  and  subtraction :  These  may  be  performed  by  the  compasses. 


A            a 
a 


t-T^ 


d   b   i 


AC-a+6 
Fig.  1. 


AC^a-b 
Fig.  2. 


(b)  Multiplication:  On  the  sides  of  an  angle  lay  off  Of/ =  1,  UA=a, 
OB  =  b;  join  BtoU  and  draw  AC  through  A  parallel  to  BU.   Then  BC  =  ab. 


(c)  Division:  Onjthe  sides  of  an  angle  take  0C/  =  1,  OB  =  b,  BA  =  a; 
join  B  to  U  and  draw  AC  through  A  parallel  to  BU.    Then  UC  =  a/6. 


(d)  Extraction  of  square  root:  Take  AB  =  a,  [BC  =  1,  describe  a  semi- 
circle on  AC  as  a  diameter,  and  erect  BD  J-  AC.    Then  BD  =  Va. 


XX     SOME  PRINCIPLES  OF  ALGEBRA  AND  GEOMETRY 

It  follows  that  an  unknown  line  x  can  be  constructed  by  ruler  and  com- 
passes if  X  can  be  expressed  in  terms  of  known  lines  a,  6,  c,  etc.,  by  means 
of  addition,  subtraction,  multiplication,  division,  and  the  extraction  of 
square  roots. 

Unless  X  can  be  so  expressed,  it  cannot  be  constructed  by  ruler  and 
compasses.  This  is  proved  by  a  combination  of  algebra  and  analytic 
geometry,  and  is  used  to  prove  that  some  problems  of  construction,  for 
example,  the  trisection  of  an  angle  or  the  squaring  of  a  circle,  cannot  be 
solved  with  ruler  and  compasses. 

17.   Some  letters  from  the  Greek  alphabet. 


a 

Alpha 

M 

Mu 

^ 

Beta 

TT 

Pi 

7 

Gamma 

S,  cr 

Sigma 

A,  5 

Delta 

0 

Phi 

Q 

Theta 

^ 

Psi 

CO 

Omega 

ELEMENTARY   FUNCTIONS 

CHAPTER  I 

FUNCTIONS,  EQUATIONS,  AND   GRAPHS 

1.  Comparison  of  the  Reasoning  in.  Natural  Science  and  in 
Mathematics.  In  science,  the  method  of  procedure  in  deter- 
mining the  law  of  phenomena  is  as  follows: 

1.  Observations  are  recorded,  compared,  and  classified. 

2.  An  induction  is  made  and  the  generahzation  resulting 

is  stated  as  a  hypothesis. 

3.  Deduction  from  the  hypothesis  leads  to  a  conclusion. 

4.  The  conclusions  are  tested  and  verified  by  experiment. 

Mathematical  reasoning  is  not  inductive,  and  hence  it  is  not 
of  the  nature  of  stages  1  and  2  of  the  scientific  method.  But 
when  a  science  has  advanced  to  the  point  where  the  data  are 
expressed  in  terms  of  magnitude,  the  generalization  can  be 
expressed  mathematically  in  simple  and  compact  form,  and 
the  deductive  process  of  the  scientific  method  can  be  carried 
out  by  the  direct  and  powerful  methods  of  mathematics. 

The  reasoning  in  mathematics  is  purely  deductive  in  charac- 
ter, and  the  conclusion  reached  contains  no  more  than  the 
hypothesis  from  which  it  was  derived.  If  the  conclusion 
were  more  general  than  the  hypothesis,  then  it  would  be 
certain  that  the  deductive  reasoning  was  not  performed  cor- 
rectly. While  a  correctly  deduced  conclusion  states  nothing 
which  was  not  included  in  the  hypothesis,  it  is  in  a  form 
which  can  be  more  easily  comprehended  and  more  readily 
used.  The  process  of  going  from  hypothesis  to  conclusion  may 
be  likened  to  the  unwrapping  of  a  compact  bundle;  there  is 
no  more  pertaining  to  the  bundle  at  the  end  of  the  operation 

1 


2  ELEMENTARY  FUNCTIONS 

than 'there  was  at  the  beginning,  but  the  contents  are  dis- 
closed to  the  mind,  and  can  be  examined.  What  is  impUcit 
in  a  hypothesis  becomes  exphcit  in  the  conclusion. 

In  natural  science,  if  the  conclusion  when  tested  does  not 
check  with  experience,  the  observations  are  reexamined,  in- 
creased in  number,  and  in  some  cases  after  a  long  interval  of 
time  and  much  work  by  many  men,  a  new  hypothesis  is  stated, 
and  the  rest  of  the  scientific  process  repeated.  In  mathematics, 
the  question  of  the  truth  or  usefulness  of  the  hypothesis  may 
not  he  raised,  but  the  accuracy  of  the  deduction  is  an  ever- 
present  concern,  and  should  he  constantly  tested  hy  checks. 

The  verification  in  the  scientific  method  involves  reference 
to  the  field  of  observation  and  is  not  deductive  in  character, 
and  hence  not  mathematical. 

2.  Example  of  the  Utility  of  Mathematics  in  Science.  A 
classical  example  of  the  scientific  method  and  the  part  which 
mathematics  plays  in  natural  science  is  furnished  by  the  steps 
leading  to  the  discovery  of  the  planet  Neptune. 

Observations  on  the  motions  of  the  planets  of  the  solar 
system  were  recorded  in  great  number  by  the  astronomer 
Tycho  Brahe  (1546-1601). 

His  assistant,  Johannes  Kepler  (1571-1630),  generalized 
from  the  observations  and  stated  the  hypotheses  known  as 
Kepler's  laws. 

Sir  Isaac  Newton  (1642-1727),  by  means  of  mathematics, 
condensed  Kepler's  three  laws  into  one,  the  law  of  gravitation. 

Up  to  1846  Uranus  was  the  outermost  planet  of  the  solar 
system  then  known.  The  irregularities  in  its  orbit  led  astrono- 
mers to  suspect  that  there  was  another  planet  outside  Uranus 
which  caused  these  disturbances. 

LeVerrier  and  Adams,  independently,  using  Newton's  law 
and  the  facts  of  the  disturbances,  deduced  mathematically  the 
position  that  an  outer  planet  must  occupy  to  produce  these 
perturbations.  Adams  spent  months  in  carrying  through  the 
intricate  calculations  necessary  and  in  checking  the  accuracy 
of  his  deductions,  which  he  did  by  solving  the  problem  a  number 
of  times  in  diCuerent  ways. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS      3 

The  last  stage,  the  verification,  was  performed  by  Galle, 
who  turned  his  telescope  in  the  direction  indicated  by  LeVer- 
rier,  and  on  September  23,  1846,  discovered  Neptune.  Here 
was  a  wonderful  verification  of  Newton's  law  and  a  monument 
to  the  power  and  value  of  mathematics  in  making  deductions. 

3.  Forms  in  which  Data  are  Recorded.  The  observations 
of  the  scientist  are  usually  recorded  in  the  form  of  a  table  of 
two  related  sets  of  magnitudes.  The  biologist  compares  the 
width  of  a  leaf  with  the  number  of  specimens  examined  having 
that  width;  the  physicist,  the  expansion  of  a  substance  with 
the  temperature;  the  chemist,  the  amount  of  substance  in 
solution  with  the  time. 

The  following  tables  illustrate  the  manner  of  recording  the 
data. 


Width  of  leaf  in  inches. 


Number  of  specimens. 
Temperature,  degrees  Cent. 


1.4,     1.6,     1.8,    2.0,    2.2,    2.4,     2.6 


3,         7,        9,        14, 

10,       8,         2 

0,    200,       400,       600, 

800,       1000 

0,  0.075,  0.166,  0.266, 

0.367,  0.466 

15,          30,           45, 

60,           75 

Expansion  of  porcelain  rod, 
100  mm.  long,  in  mm. 

Time  in  seconds. 

Sugar  in  solution,  grams.         |     .046,       .088,       .130,       .168,       .206 

The  data  are  sometimes  recorded  by  an  instrument.  The 
temperature  and  barometer  records  of  the  Weather  Bureau  are 
so  recorded.  A  pen  is  connected  with  the  thermometer  and 
rests  against  a  sheet  of  paper  fastened  to  a  revolving  cylinder. 
The  cylinder  revolves  steadily  and  the  pen  rising  and  falUng 
with  the  temperature  traces  a  curve  on  the  paper,  from  which 
the  temperature  at  any  time  may  be  determined. 

The  variation  in  the  tide  at  an  important  port  is  similarly  re- 
corded on  a  chart  fastened  to  a  revolving  cylinder  by  a  pen  at- 
tached to  a  float  which  rises  and  falls  with  the  tide  (Figure  6). 

The  data  may  be  presented  most  conveniently  in  the  form  of 
a  generahzation,  expressed  either  in  words  or  in  mathematical 
symbols.  For  example,  a  comparison  of  the  pairs  of  values 
in  the  second  table  above,  neglecting  the  first  two  pairs  of  values, 
leads  to  the  following  generalization,  expressed  in  words: 


ELEMENTARY  FUNCTIONS 


For  a  temperature  of  400°,  or  greater,  an  increase  of  200°  in 
the  temperature  causes  an  expansion  of  approximately  0.1  of  a 
millimeter  in  the  length  of  the  rod. 

The  generahzation  may  also  be  expressed  in  symbols.  If 
every  increase  of  200°  causes  an  expansion  of  0.1  of  a  millimeter, 


SUiHon                      III 

LiiTuir  Hours  after  Moon' a  Upper  Transit 
VI                IX              XII               XV           XV III          XXI 

^:s_       

Ni 

-      -           --          ^      S           - __^_      _ 

i 

:  :     :    ^        S:  :   ::  "  --  -:        - 

Boston                  \ 

Z                S                             J     '  ~    ~ 

Mass.                     -L 

-  -    zi              ~_s-  ::  ::  ":  J —  - 

J.                            S                         T 

5 

~    ~    ~    /    -"Jl~          ---^--    --    --2 

/''         ~                                              ""^s             ^^                       ~     ' 

is 

vj_  •" 

y  'T"  ^ 

2                "^ 

V                                  -                                            ^^"'^ 

y 

N                                                            Z               S 

-V            : 

S                                     7            \ 

::   _   _     - 

o         ,               / 

^         .          -J  -      .      \       : 

-^-^     -              z   _        .          . 

^        -          ^^  -              \      -     -  -  - 

\       -          7£.                     a:            ji 

N              1                      >               /^ 

^                     -_y__  ^ 

-^     -            4                                                       5                      2" 

^         ^^            -                  it  \          ^^ 

1          Tr^-m                          IrU-Lr 

lOfl. 


5ft. 


Fig.  6. 

then  an  increase  of  1°  would  cause  an  expansion  of  0.1/200,  or 
0.0005;  hence  for  an  increase  of  temperature  of  t  degrees  the 
expansion  e  would  be 

e  =  0.0005^. 

This  equation  expresses  the  generahzation  in  more  compact 
form  than  the  sentence  above.  In  this  illustration  we  have 
considered  only  a  part  of  the  table,  a  part  for  which  the  generah- 
zation is  very  simple. 

The  determination  of  the  mathematical  expression  of  the 
generahzation  from  a  table  of  values  will  be  one  of  the  objects 
of  this  course.  The  generalization  from  a  mathematical  point 
of  view  is  considered  in  the  following  section. 

4.  Variable.  Function.  The  following  table  gives  the  lengths 
of  an  iron  bar  suspended  from  one  end  when  carrjdng  different 
loads. 


Load  in  lbs. 


Length  in  in. 


0, 


500, 


1000,         1500, 


2000 


30,      30.67,      30.91,      31.23,        31.52 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS      5 

Any  one  of  the  numbers  in  each  of  these  sets  of  numbers  is 
conveniently  represented  by  a  single  symbol,  as  x,  y,  t,  etc.  Thus 
X  may  be  taken  to  represent  one  of  the  numbers  0,  500,  1000, 
1500,  2000  and  any  other  number  that  might  be  included  in  the 
half  of  the  table  giving  loads,  while  y  may  represent  the  cor- 
responding number  giving  lengths  in  the  second  part  of  the 
table. 

The  symbols  x  and  y  are  called  variables,  in  accordance  with 
the 

Definition.  A  variable  is  a  symbol  for  any  one  of  a  set  of 
numbers. 

In  the  experiment  giving  rise  to  the  above  table,  the  load 
was  changed  arbitrarily  by  the  experimenter  and  the  length  of 
the  bar  for  the  chosen  load  measured.  In  consequence  of  this 
order  of  measurement  the  variable  representing  the  load  is 
called  the  independent  variable  and  the  one  representing  the 
length  of  the  bar  the  dependent  variable. 

It  is  customary  to  denote  the  independent  variable  by  x 
and  the  dependent  variable  by  y. 

If  the  experiment  were  repeated,  under  the  same  conditions, 
it  would  be  found  that  for  a  specified  load  the  length  of  the  bar 
would  be  the  same,  that  is,  there  is  a  law  connecting  the 
load  and  the  length  of  the  bar.  This  relation  between  the  vari- 
ables is  expressed  by  saying  that  the  length,  y,  is  a  function  of 
the  load,  x,  in  accordance  with  the 

Definition.  A  function  is  a  variable  so  related  to  another 
variable  (called  the  independent  variable)  that  for  every  ad- 
missible value  of  the  independent  variable,  one  or  more  values 
of  the  function  are  determined.  The  function  is  also  called 
the  dependent  variable. 

The  idea  of  a  function  arises  wherever  there  is  a  relation  be- 
tween magnitudes  which  are  changing,  and  it  underlies  all 
magnitude  relations  which  mankind  has  discovered. 

Example  1.  The  algebraic  expression  2x  +  3  is  a  variable  whose  value 
is  determined  whenever  a  definite  value  is  assigned  to  x.  If  x  be  given  the 
value  1,  then  2x  +  3  has  the  value  of  5,  and  if  x  has  the  value  2,  2x  +  3  haa 
the  value  7.    Hence  2a;  +  3  is  a  function  of  x. 


6  ELEMENTARY  FUNCTIONS 

Example  2.  A  theorem  from  physics  relating  to  a  falling  body  states 
that  if  a  heavy  object  be  dropped,  the  distance  it  falls  in  t  seconds  is  16^^. 
This  distance  is  a  function  of  the  variable  t;  for  if  t  be  given  a  definite 
positive  value,  the  distance  is  determined.  li  t  =  2  seconds,  the  distance 
fallen  is  64  feet;  if  <  =  3  seconds,  the  distance  is  144  feet.  Negative 
values  of  t  are  meaningless  and  hence  not  admissible. 

Example  3.  The  formula  for  the  area  of  a  circle  is  A  =  irr^.  If  the 
radius  r  be  given  a  definite  value  the  area  is  determined.  Hence  the  area 
A  is  a  function  of  r.  The  symbol  tt  represents  the  number  3.14159  . . . 
which  remains  the  same  for  any  pair  of  corresponding  values  of  r  and  A, 
and  hence  is  called  a  constant  in  accordance  with  the 

Definition.  A  constant  is  a  symbol  for  a  particular  number. 
It  has  the  same  value  throughout  a  discussion. 

Example  4.    A  simple  equation  which  occurs  frequently  in  practice  is 

y  =  mx, 

where  x  represents  the  independent  variable,  y  the  function,  and  m  is  a 
constant  representing  a  fixed  value  as  x  and  y  vary. 

If  X  represents  the  number  of  pairs  of  shoes  of  a  certain  kind  sold  during 
a  limited  time  by  a  dealer,  and  y  the  amount  of  the  sales,  then  m  represents 
the  price  per  pair  which  remains  fixed,  for  the  time  considered,  while  x  and 
2/ vary. 

Example  5.  An  equation  in  two  variables  establishes  a  functional  re- 
lation between  the  variables.  For  if  a  value  be  given  to  either,  the  cor- 
responding value  or  values  of  the  other  may  be  found  by  substituting  the 
given  value  of  one  variable  and  solving  for  the  other. 

Either  variable  may  be  regarded  as  a  function  of  the  other,  and  the  form 
of  the  function  may  be  found  by  solving  for  one  variable  in  terms  of  the 
other. 

Thus,  if  the  equation  ^y-x^^O  be  solved  for  y  and  then  for  x  we 

have 

a?  /- 

y  =  j        and       a;  =  *  2Vy- 

The  given  equation  defines  t/  as  a  function  of  x  to  be  the  function  x^/4,  and 
X  as  a  function  of  y  to  be  the  function  ±2V2/- 

Example  6.  Some  of  the  elements  entering  into  the  cost  of  a  suit  of 
clothes  are  the  supply  of  cloth,  the  supply  of  labor,  rent,  style,  etc.  As 
these  elements  vary  the  cost  of  the  suit  will  vary,  so  that  the  cost  of  the 
suit  is  a  function  of  a  number  of  variables. 

Considering  one  of  the  independent  variables  at  a  time,  a  part  of  the 
cost  of  the  suit  may  be  expressed  as  a  function  of  this  variable,  e.g.,  the 
supply  of  cloth. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS      7 

The  law  of  supply  and  demand  from  economics  states  that  the  price 
of  an  article  increases  or  diminishes  as  the  supply  diminishes  or  in- 
creases. 

If  X  represents  the  supply  of  cloth  and  y  the  price  of  the  suit  of  clothes, 
then  it  is  assumed  in  economics  that  the  functional  relation  may  be  ex- 
pressed by  the  equation 

m 

where  m  is  a  constant  which  can  be  determined  in  any  concrete  case. 

In  this  course  we  shall  confine  ourselves  to  the  study  of 
functions  of  one  variable. 

Example  7.  The  temperature  at  a  given  place  is  a  function  of  the  time. 
For  at  a  given  time  the  temperature  must  have  a  definite  value.  But  this 
function  is  so  little  understood  that  the  Weather  Bureau  can  only  approxi- 
mate the  value  for  any  future  time,  and  that,  indeed,  only  for  times  in  the 
hnmediate  future. 

The  data  of  such  departments  of  human  knowledge  as  physics, 
astronomy,  and  engineering  are  so  complete  that  many  of  the 
functions  arising  there  can  be  identified  and  studied  by  mathe- 
matical methods.  In  other  subjects,  for  example,  chemistry 
and  economics,  the  data  have  only  recently  been  made  suf- 
ficiently complete  to  warrant  an  increasing  use  of  mathematics. 
But  there  still  remains  a  countless  number  of  functions  which 
mankind  has  been  unable  to  represent  by  a  mathematical 
expression. 

EXERCISES 

In  the  following  exercises  give  the  reason  for  the  statement  that  one 
variable  is  a  function  of  another. 

1.  Mention  three  variables  which  are  functions  of  the  side  of  an  equi- 
lateral triangle  of  varying  size. 

2.  A  train  goes  from  one  station  to  another  at  a  variable  rate.  Mention 
two  variables  of  which  the  rate  is  a  function. 

3.  What  are  some  of  the  variables  of  which  the  cost  of  erecting  an  office 
building  is  a  function? 

4.  Mention  some  variables  involved  in  heating  water  in  a  pan  on  a  gas 
stove.     Which  are  independent  variables?    Which  are  functions? 

5.  Find  the  functions  of  x  defined  by  the  following  equations  and  tabu- 
late three  pairs  of  values. 


8  ELEMENTARY  FUNCTIONS 

(a)  2/  -  x2  -  3x  -  2  =  0.  (b)  x^  -  3?/2  +  2^/  +  4  =  0. 

(c)  3xy  +  Qx  -9y  +  ^  =  0.  (d)  2x^  +  ^xy  +  4y^ +  Qx  -  Sy +  7  =  0 

5.  Notation  for  a  Function.  It  is  convenient  to  represent 
a  function  of  the  variable  x  by  the  symbol  f{x)  which  is  read 
"  function  of  x,"  *^  the  function  /  of  a;,"  or  merely  ^'  f  of  a;." 
The  various  parts  of  the  symbol  are  to  be  regarded  as  forming 
a  single  compound  symbol,  never  as  separate  symbols  meaning 
the  product  of  two  numbers  /  and  x. 

This  symbol  is  used  to  denote  either  any  function  or  a  par- 
ticular function  such  as 

fix)  =x^  +  x-l. 

Similar  symbols  convenient  for  distinguishing  different 
functions  are 

F(^),  g(.x),  </>W,  etc. 

An  advantage  of  this  notation  is  that  the  value  of  a  function 
f(x)  for  any  value  of  x,  say  x  ^  a,  may  be  suggestively  represented 
hy  f{a).     For  example,  if 

f{x)  =  x2  +  a;  -  1, 

/(a)  =  a^  +  a  -  1,  /(2)  =  2^  +  2  -  1  =  5, 

/(-I)  =  (-1)2  +  (-1)  -  1  =  -  l,/(0)  =  -  1, 

f{-x)  =  {-xY  +  i-x)  -  1  =  a:2  -  X  -  1,  etc. 

This  notation  also  enables  us  to  state  certain  theorems  in  a 
more  compact  form. 

EXERCISES 

1.  If  /(x)  =  x3  _  xS  find  /(I),  /(  -  3),  /(O),  /(-  X). 

2.  If  F{x)  =  l/x2,  find  F(2),  F{-  1),  F{a),  F{-  x). 

3.  If  </)(x)  =  mx  +  6,  find  </)(0),  0(1),  (^(xj),  0(-  6/m). 

4.  If  /(x)  =  x3  +  x,  show  that/(-  2)  =  -/(2),  that/(-x)  =  _/(x). 

6.  If  /(x)  =  x«  +  x2  show  that  /( -  2)  =  /(2),  that  /( -  x)  =  /(x) . 

6.   If  /(x)  =  X*  +  X  +  1,  determine  whether  either  /(-  x)  =  /(x)  or 
/(-x)  =  -/(x)  is  a  true  relation. 

6.  Determination  of  the  Function  which  Expresses  the 
Functional  Relation  between  Two  Variables.  The  functional 
relation  between  two  variables  is  expressed  in  s5mabols  when- 
ever possible,  for  the  sake  of  the  greater  simplicity  which  this 


^^ 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


9 


gives  to  the  generalization  and  of  the  greater  ease  in  making 
deductions.  If  x  is  chosen  to  represent  the  independent  vari- 
able, then  an  algebraic  expression  is  sought  to  represent  the 
function. 


Example  1.     If  x  represents  any  one  of  the  numbers  in  the  first  column 
f(x)      of  the  accompanying  table,  or  the  value  of  the  independent 

0  variable,  what  function  of  x  will  represent  the  second  column? 

1  We  see  by  inspection  that  each  number  in  the  second  set  is 
4        the  square  of  the  corresponding  number  of  the  first  set. 
9            Hence  /(x)  =  x^  is  the  required  function. 

Check.    When  x  =  0,/(0)  =  0;  when  x  =  !,/(!)  -  1,  etc.,  for  all  the  table. 

Example  2.  Find  the  area  of  an  equilateral 
triangle  as  a  function  of  the  side. 

Let  X  represent  the  side  of  the  equilateral 
triangle  ABC  with  the  altitude  CD. 

Then 

AD  =  x/2     (The  altitude  bisects  the  base.) 
CD2  =  AC^  -  AD^  (ACD  is  a  right  triangle.) 


(x/2)2  =  x2  -  xV4 


Hence 


Vs      Vs 


x^,  which  is  the  required 


Therefore,  the  area  =  ^  AB  -DC  -  ^  x  • 

function  of  x. 

Example  3.  A  man  leaves  a  village  10  miles  directly  east  of  a  city  and 
walks  east  at  the  rate  of  3  miles  an  hour.  Express  the  distance  he  is  from 
the  city  at  any  time,  t,  after  he  starts  as  a  function  of  t.  Where  will  he 
be  at  the  end  of  3  hours,  and  when  will  he  be  25  miles  from  the  city? 

Since  distance  equals  the  rate  multipUed  by  the  time,  the 
distance  the  man  is  from  the  village  at  any  time  t  after 
he  starts  is  St;    and  the  distance  he  is  from  the  city  is 
s  =  St  +  10,  since  the  village  and  city  are  10  miles  apart 
and  he  goes  away  from  both. 

Hence  the  required  function  is  3t  +  10. 

When  «  =  3,  s  =  3 -3  +  10  =  19;  when  s  =  25,  25  =  3«  +  10,  whence  t  =  5 

The  data  and  results  are  recorded  in  the  accompanying  table. 

Example  4.     A  ball  rolls  down  an  inclined  plane.    The  distance  s  it 

rolls  in  the  lib.  second  is  recorded  in  the  table.     Express  the  distance  the 

ball  rolls  in  any  second  as  a  function  of  t.    What  distance  will  it  roll  in 

the  5th  second? 


t 

s 

rate 

0 

10 

3 

3 

19 

5 

25 

10 


ELEMENTARY  FUNCTIONS 


t   I s_  The  values  of  both  the  mdependent  variable  and  the  function 

1  4        form  arithmetic  progressions. 

2  12  The  formula  for  the  nth  term  of  an  arithmetic  progression  is 

3  20        l  =  a+  (n  -1)  d. 

The  common  difference  d  of  the  values  of  the  function  is  8,  the  first 
term  a  is  4,  the  number  of  terms  n  is  t,  and  I  =  s 

Hence  we  have  s  =  4+(i-l)8  =  8i-4  which  is  the  required  function. 
Whent  =  5,/(5)  =  8-5  -  4  =  36. 

Check.  If  the  values  1,  2,  3  are  substituted  for  « in  8«  -  4  the  values  4, 
12,  20  are  obtained,  which  are  the  values  of  s  given  in  the  table. 

Example  5.  If  $100  are  placed  in  a  bank  at  5  %  interest,  compounded 
annually,  what  is  the  amount  at  the  beginning  of  tth  year? 

The  amount  at  the  beginning  of  the  first  year  is  of  course  $100.  The 
interest  for  the  first  year  is  $5  and  the  amount  at  the  beginning  of  the 
second  year  is  $105. 


The  interest  for  the  second  year  is  $5.25  and  the 
amount  at  the  beginning  of  the  third  year  is  $110.25. 
The  accompanying  table  gives  the  data  obtained. 
The  values  of  the  function  form  a  geometric  pro- 


=  1.05. 


100 
105 
110.25 

gression,  for  the  ratio  of  the  second  value  to  the  first  is  equal  to  the  ratio 
of  the  third  value  to  the  second,  i.e., 

105      ,  „.  ,         110.25 

100  =  ^-^^       ^^^       -W 

The  nth  term  of  a  geometric  progression  is  given  by  the  formula 

In  this  case  1  =  A,  n  =  t,  a  >=  100  and  r  =  1.05.    Substituting  these  values 
in  the  above  formula,  we  have  for  the  required  function 

A  =  100(1.05)'-!. 

Check.  Substituting  the  values  1,  2,  3,  for  t  in  this  function  we  get  the 
values  100  105  and  110.25  respectively  for  A,  which  results  agree  with  the 
table. 

EXERCISES 

1.  In  each  of  the  following  tables  let  z  represent  any  one  of  the  numbers 
in  the  first  column  (value  of  the  independent  variable),  and  find  a  function 
of  X  which  will  represent  the  corresponding  number  in  the  second  colunm. 
Add  two  additional  pairs  of  values  of  x  and  the  function  to  each  table. 

(a) 


X 

0 

(b)  X 
0 

fix) 
0 

(c)      X 
1 

fix) 

1 

(d)    x 

1 

fix) 

0 

1 

1 

1 

1 

1 

27 

3 

4 

8 

2 

8 

4 

2 

64 

4 

9 

27 

3 

27 

9 

3 

125 

5 

16 

64 

(e) 


X 

FTJNCTIONS, 

fix)           (f)    X 

EQUATIONS,  AND  GRAPHS 

fix)          (g)      a:      /(x)          (h)     X 

11 

1 

8 
27 
64 

1                   0 

4                  2 

9                  4 

16                  6 

0                      0 
3                     1 
6                      2 
9                      3 

3                     1 
5                     2 
7                     3 
9                    4 

2 

4 

8 

16 

2.  Find  the  perimeter  of  an  equilateral  triangle  as  a  function  of  the 
altitude  x. 

3.  Express  the  area  of  a  square  as  a  function  of  (a)  the  side,  (b)  the 
diagonal. 

4.  Express  the  area  of  a  regular  hexagon  as  a  function  of  the  side. 

5.  A  regular  octagon  is  formed  by  cutting  off  isosceles  right  triangles 
from  the  corners  of  a  square.  Express  the  area  as  a  function  of  the  side 
of  one  of  the  right  triangles  cut  off. 

■6.  Express  the  side  of  a  regular  decagon  as  a  function  of  the  radius  of 
the  circumscribed  circle. 

7.  A  Norman  window  consists  of  a  square  surmounted  by  a  semi- 
circle.    Find  the  area  as  a  function  of  the  side  of  the  square  x. 

8.  (a)  A  man  walks  from  a  certain  town  toward  a  second  at  the  rate  of 
three  miles  an  hour.  Express  the  distance  traveled  as  a  function  of  the 
time. 

(b)  A  second  man  starts  at  the  same  time  from  the  second  town,  which 
is  10  miles  from  the  first,  and  travels  at  the  rate  of  4  miles  an  hour  toward 
the  first  town.  Express  his  distance  from  the  first  town  as  a  function  of 
the  time. 

(c)  A  third  man  starts  from  the  same  town  as  the  first  man  but  two 
hours  later  and  travels  at  the  rate  of  3^  miles  an  hour  in  the  same  direction. 
Express  his  distance  from  town  as  a  function  of  the  time  elapsed  since  the 
first  man  started. 

How  far  will  each  man  be  from  the  first  town  4  hours  after  the  fiirst  man 
starts?     When  will  each  man  be  20  miles  from  the  first  town? 

9.  A  man  starts  from  a  town  15  miles  directly  west  of  a  city  and  travels 
east  at  the  rate  of  4  miles  an  hour.  A  second  man  starts  from  the  same 
town  at  the  same  time  and  travels  west  at  the  rate  of  3  miles  an  hour. 
Express  the  distance  of  each  from  the  city  after  t  hours  as  a  function  of  t. 
How  far  from  the  city  will  each  be  in  3  hours?  When  will  they  be  25 
miles  apart? 

10.  A  ball  starting  from  rest  rolls  down  an  inclined  plane  4  feet  in  the 
first  second,  8  feet  in  the  next,  12  feet  in  the  next,  etc.  Express  the  dis- 
tance it  rolls  in  any  second  t  as  a  function  of  t.  How  far  will  it  roll  in  the 
8th  second? 

11.  With  the  data  of  the  preceding  problem  and  the  formula  for  the  smn 
of  an  arithmetic  progression  find  the  total  distance  rolled  as  a  function  of 
the  time  t.    When  will  the  ball  have  rolled  108  feet? 


\ 
12  ELEMENTAKY  FITNCTIONS 

12.  A  body  starting  from  rest  falls  16  feet  in  the  first  second,  48  feet  in 
the  next,  80  feet  in  the  next,  etc.  Find  the  distance  fallen  in  any  second 
as  a  function  of  the  time  of  falling.  Find  the  total  distance  fallen  as  a 
function  of  the  time. 

13.  $100  is  placed  at  simple  interest  at  4  per  cent.  Express  the  amount 
at  the  end  of  t  years  as  a  function  of  t. 

14.  If  the  interest  in  the  preceding  problem  is  compounded  annually, 
express  the  amount  at  the  end  of  t  years  as  a  function  of  t. 

15.  The  cross-section  of  a  gutter  pipe  is  in  the  form  of  an  isosceles 
trapezoid.  The  lower  base  and  the  inclined  sides  are  each  3  inches  long. 
Find  the  area  as  a  function  of  the  width  across  the  top. 

16.  A  rectangle  is  inscribed  in  a  circle  4  inches  in  diameter.  Express 
the  area  as  a  function  of  one  of  the  sides. 

7.  Graphical  Representation.  Directed  Lines.  The  func- 
tional relation  can  be  represented  by  a  geometrical  figure  which 
fui-nishes  a  valuable  method  for  studying  the  properties  of  the 
function,  since  the  whole  of  the  relation  is  placed  before  the 
mind  at  once.  The  system  of  coordinates  devised  by  Rene 
Descartes  (1596-1650),  which  is  developed  in  the  following 
section,  is  the  basis  of  the  representation.  This  system  of 
coordinates  rests  on  the  theory  of  directed  lines. 

Let  XX'  be  any  line,  and  let  the  direction  from  X'  to  X  be 
called  positive,  from  X  to  X'  negative.  These  words  are  used 
instead  of  such  terms  as  north  and  south,  to  the  right  and  left, 
up  and  down,  backward  and  forward.  A  line  upon  which  a 
positive  direction  has  been  fixed  is  called  a  directed  line.  The 
positive  direction  is  commonly  indicated  by  an  arrow-head. 

If  A  and  B  be  two  points  on  a  directed  line,  the  symbol  AB 
is  used  to  denote  either: 

(1)  The  Hne  drawn  from  ^4  to  B,  or 

(2)  The  real  number  whose  numerical  value  is  the  number  of 
times  the  unit  of  length  is  contained  in  the  line,  and  whose 
sign  is  positive  or  negative  according  as  the  direction  from  A 
to  B  is  positive  or  negative. 

Thus,  in  the  figure,  AB  and  A'B'  denote  certain  lines. 

x'\ — \ — f— HH — hH — \—\ — I— HH — I— f-x 

A  B  B'  A' 

Fig.  8. 


^      FUNCTIONS,  EQUATIONS,  AND  GRAPHS     13 

They   also   denote   the   numbers  AB  =  S   and   A'B'  =  —  2, 
provided  the  unit  of  length  is  a  quarter  of  an  inch. 
It  follows  that  if  A  and  B  are  points  on  a  directed  line 


BA  =-.  -AB  (1) 


I 

B  Definition.     No  matter  what  the  position  of  three  points 

A,  B,  C,  on  SL  directed  Une  may  be,  the  sum  oi  AB  and  BC  is 

defined  to  be 

AB  +  BC  '=  AC  (2) 

If  ^5  is  thought  of  as  a  motion  from  A  to  B,  and  BC  as  a 
motion  from  B  to  C,  the  sum  gives  a  motion  from  A  to  C. 
The  sum  gives  the  distance  from  A  to  C  in  both  magnitude  and 
direction,  but  not  necessarily  the  total  number  of  units  passed 
over  in  going  from  A  to  B  and  then  from  B  to  C. 

That  this  definition  agrees  with  elementary  algebra,  when 
AB,  BC,  and  AC  are  regarded  as  numbers,  is  seen  in  the  fol- 
lowing illustrations,  of  which  the  first  agrees  with  arithmetic. 

xi    I  ^  I    I    I   ?  I   y  I    l>x  XI    I  -t    I    ^  I    I    I    I   f  I    |.x 

AB+ BC  =  4^  +  2  =  G=  AC  AB+BC=? 

yi  f   I    [  f   i    I    I    [   ?  t>X  X'\     I    ?   I    1    I 


AB+BC=?  AB  + BC=-S+i-'i)  =  -7  =  AC 

X'l     I     ?    I     I     I    ^     I     f    I      I      |.X  X^l    ?     I      I     I     I     ?     I     I    ^    I     |>X 

AB+BC  =  !  AB  +  BC=1 

Fig.  9. 

An  especially  important  use  of  directed  lines  is  the  following: 
Let  0  be  any  point  on  a  directed  line  X'  X,  and  let  points  be 

laid  off  on  each  side  of  0  at  a  unit's  distance  from  each  other. 

Then  with  every  point  P  on  X'X  is  associated  a  real  number  OP, 

and  conversely,  with  every  real  number  is  associated  a  point  on 

the  line. 


X'- 


I   ol     I      I     I      I     I 


-^x 


Fig.  10. 


In  the  figure,  the  numbers  associated  with  the  points  A,  B, 
C,  D  are  respectively  OA  =  3,  OB  -  -  3,  OC  ^  5,  OD  =  -4.5. 


u 


ELEMENTARY  FUNCTIONS 


Using  this  association  of  points  and  numbers,  if  Pi  and  Pi 
are  two  points  on  the  Hne,  and  xi  and  X2  are  the  numbers  asso- 
ciated with  them,  we  have  from  (2) 

OPl+PlP2   =  OP2, 

^1 


JC' 


I    I    I    I    I    I    r 


■^x 


Fig.  11. 

whence 

Pi  Pa  =  OP2  -  OPi  =  X2  -  xi  (3) 

This  difference  of  the  values  of  the  x^s  is  denoted  by  Ax,  so 
that 

Ax  =PiP2=X2-Xi  (4) 

Notice  that  Ax  is  a  single  symbol  (never  the  product  of  two 
numbers  A  and  x),  and  that  its  value  is  obtained  by  subtract- 
ing the  value  of  x  corresponding  to  the  first  point  from  that 
corresponding  to  the  second. 


EXERCISES 

1.  Illustrate  (3)  by  numerical  examples  for  the  six  possible  relative 
positions  of  0,  Pi,  and  P2.     Find  Ax  for  each  case. 

2.  Show  that  Ax  is  positive  or  negative  according  as  Pi  lies  to  the  left 
or  right  of  Pi,  using  the  definition  that  a<feif6-ais  positive. 

3.  If  OP  =  X,  show  that  x  increases  or  decreases  according  as  P  moves 
to  the  right  or  left. 

8.  Rectangular  Coordinates.    Let    X'X  and  Y' Y  be  two 

perpendicular  directed  Hues  in- 
tersecting at  0.  Let  the  posi- 
tive direction  on  X'  X  and  on 
all  lines  parallel  to  it  be  to  the 
right,  and  let  that  on  Y'  Y, 
and  on  all  Hues  parallel  to  it, 
be  upward. 

Let  P  be  any  point  in  the 
plane,  and  draw  PM  ±  X'X, 
and   PN±  Y'Y.      Then  the 
numbers  OM  =  x  and  ON  =  y 
are  called  rectangular  coordinates  of  P,  x  the  abscissa,  and  y  the 


X' 


Y 

r 

M     . 

0 

N 

D 

y" 

FiQ.  12. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     15 

ordinate.  In  the  figure,  the  ordinate  y  is  often  thought  of  as 
MP,  which  equals  OiV  in  both  magnitude  and  direction.  The 
directed  Unes  X'  X  and  F'  Y  are  called  the  axes  of  coordinates 
and  their  intersection  0  the  origin. 

The  abscissa  of  a  point  is  positive  or  negative  according  as 
the  point  lies  to  the  right  or  left  of  the  2/-axis.  The  ordinate 
is  positive  or  negative  according  as  the  point  hes  above  or 
below  the  a;-axis.  In  the  figure,  the  abscissa  x  is  positive, 
while  the  ordinate  y  is  negative. 

As  seen  above,  any  point  P  determines  a  pair  of  real  num- 
bers, its  coordinates.  Conversely,  given  any  pair  of  real  num- 
bers, X  and  y,  a  point  may  be  plotted,  that  is,  constructed,  whose 
coordinates  are  x  and  y.  For  on  X' X  lay  off  OM  =  x.  At 
M  erect  a  line  perpendicular  to  X' X  and  on  it  lay  off  MP  =  y. 
Then  P  is  the  required  point. 

The  symbol  (x,  y)  is  used  to  mean  the  point  whose  coordinates 
are  x  and  y.  If  P  is  this  point,  it  is  indicated  by  the  sjrmbol 
P{x,  y). 

Coordinate  axes  divide  the  plane  into  four  parts  called 
quadrants.  These  are  numbered  as  in  the  figure,  which  also 
indicates  the  signs  of  the  coordi-  yk 

nates  of  a  point  in  every  quadrant. 

We    are    frequently    concerned  ,      ^  . 

with  points  which  are  symmetric 
with  respect  to  the  origin,  the  axes, 


or  a  fine  bisecting  the  first  and     Xf 

third    quadrants.     Points   ha\dng 

these  sjmametric  relations  are  de-  iii(— ,-)         iv(+i-) 

termined  in  accordance  with  the 

Definitions.     (1)  Two   points  ^ 

are  symmetric  with  respect  to  a  ^^' 

third  point  if  the  line  joining  the  two  points  is  bisected  at  the 
third  point. 

(2)  Two  points  are  symmetric  with  respect  to  a  line,  if  the 
line  is  the  perpendicular  bisector  of  the  fine  joining  the  two 
points. 


16  ELEMENTARY  FUNCTIONS 


EXERCISES 


1.  Plot  the  points  whose  coordinates  are  given  below,  and  determine 
the  nature  of  the  symmetry  for  the  pairs  of  points  in  each  group. 

(a)  (2,1),   (-2,1);  (3,0),   (-3,0);  (-2,-3),   (2,-3). 

(b)  (2,  1),    (2,  -  1);   (0,  5),   (0,  -  5);  (3,  -  4),   (3,  4). 

(c)  (2,1),    (-2,-1);   (-3,1),   (3,-1);  (4,-1),   (-4,1). 

(d)  (2,1),    (1,2);   (1,3),   (3,1);  (3,-5),   (-5,3). 

2.  By  means  of  the  corresponding  parts  of  Exercise  1,  what  can  be  said 
of  the  positions  of  the  following  pairs  of  points? 

(a)   {x,  y)  and  (-  x,  ?/).  (b)   {x,  y)  and  (x,  -  y). 

(c)   {x,  y)  and  (-  x,  -y).  (d)  (x,  y)  and  {y,  x). 

3.  One  end  of  a  line  bisected  by  the  origin  is  the  point  (-  5,  2).  What 
are  the  coordinates  of  the  other  end? 

4.  What  are  the  coordinates  of  the  point  symmetrical  to  the  point 
(—3,  4)  with  respect  to  the  y-axis?  the  x-axis?  the  origin?  the  line 
bisecting  the  first  and  third  quadrants? 

5.  Find  the  coordinates  of  the  vertex  or  vertices  not  given  in  the  regular 
polygons  located  as  follows: 

(a)  One  vertex  of  an  equilateral  triangle  is  the  point  (1,  0)  and  the 
altitude  through  this  vertex,  which  is  v3  units  long,  extends  through  the 
origin. 

(b)  An  equilateral  triangle  has  vertices  with  coordinates  (0,  0)  and  (1,  0). 

(c)  A  square  with  opposite  vertices  having  coordinates  (1,  0)  and 
(-  1,  0). 

(d)  A  hexagon  two  of  whose  opposite  vertices  have  coordinates  (1,  0) 
and  (-1,0). 

(e)  An  octagon  with  two  opposite  vertices  having  coordinates  (1,  0) 
and  (-  1,  0). 

6.  The  coordinates  of  three  vertices  of  a  rhombus  are  (-  1,  0),  (0,  V3), 
(1,  0).  What  are  the  coordinates  of  the  fourth  vertex?  (Three  solutions.) 
What  are  the  coordinates  of  the  intersection  of  the  diagonals? 

9.  Graph  of  a  Function.  Values  of  x  and  the  corresponding 
values  of  a  function  may  be  exhibited  in  tabular  form. 

Table  1  gives  the  population  of  the  United  States  in  millions 
for  the  successive  decades  from  1830  to  1910. 

rp  , ,    .      x_\  1830,    1840,    1850,    1860,    1870,    1880,    1890,    1900,    1910, 
^^    '    J/  I  12.8,      17,     23.1,   31.4,   38.5,   50.1,    62.6,    75.9,    93.9, 

Table  2  gives  pairs  of  values  of  x  and  the  function  J  x  +  f  • 

Table  2  x       1-7,        -5,        -2,        -1,  0,       1,       3,         5. 

laoiez.    ^a-^i     I     _i,  0,       1.5,  2,       2.5,       3,       4,        5, 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


17 


V 

-80 
-70 
-€0 
-50 
-iO 
-SO 

-go 

-10 

/ 

/ 

J 

Y 

/ 

y 

\y 

_^ 

^ 

— -j 

r-"^ 

x^soisfo  moo  laco  jn^o  mao  uw  urn  mo'Jc\ 

Fig.  14. 


Any  such  table  of  values  may  be  strikingly  exhibited  to  the  eye 
by  plotting  the  points  whose  coordinates  are  the  pairs  of  num- 
bers in  the  table,  and  then  drawing  a  smooth  curve  through  the 
points  so  obtained.    Proceeding 
thus,  we  obtain  from  Tables  1 
and  2  the  curves  in  Figm-es  14 
and    15,  which   are   called  the 
graphs  of  the  functions. 

Definition.  The  graph  of  a 
function  is  a  curve  such  that 
(1)  Any  point  whose  coordinates 
are  corresponding  values  of  x 
and  the  function  is  on  the  curve, 
and  (2)  Conversely,  the  coordi- 
nates of  any  point  on  the  curve 

are  a  pair  of  corresponding  values  of  x  and  the  function. 
Hence  the  important  fact: 

The  ordinate  of  any  point  on 
the  graph  represents  the  value  of 
the  function  when  x  equals  the 
abscissa  of  the  point. 

In  the  above  definition  the 
word  curve  is  used  in  a  very  gen- 
eral sense  to  mean  one  or  more 
lines,  straight  or  curved,  or  parts 
of  such  Hues.  Functions  exist 
which  have  no  graphs,  and  the  graphs  of  others  are  merely  one  or 
more  isolated  points,  but  we  shall  not  encounter  them  in  this 
course. 

The  graph  of  a  function  may  be  constructed  by  the  following 
process : 

Construct  a  table  of  values  of  x  and  the  function  of  x. 
Plot  the  points  whose  coordinates  are  the  pairs  of  numbers  in 
this  table. 
Draw  a  smooth  curve  through  these  points. 
In  constructing  a  graph,  notice  that  values  of  x  giving  im- 
'i  aginary  values  of  the  function   are  discarded,  and  that  the 


V 

^ 

^ 

g 

^ 

X 

f-i 

X 

^ 

L^ 

■I 

-- 

^ 

^ 

X' 

y^ 

X 

-. 

-J 

'  -J 

0 

i 

; 

i 

. 

X 

_J 

1 

_ 

_ 

Fig.  15. 


18 


ELEMENTARY  FUNCTIONS 


number  of  points  plotted  must  be  large  enough  to  indicate 
without  doubt  the  form  of  the  curve.  Whenever  it  is  not  clear 
just  how  the  ciu-ve  is  to  be  drawn,  enlarge  the  table  of  values, 
either  by  giving  more  integral  values  of  x,  or  by  assuming  for  x 
intermediate  values  such  as  2.5,  2.8,  etc.,  as  may  best  serve 
the  purpose.  The  necessity  for  this  last  remark  is  shown  by 
the  fact  that  three  points,  situated  as  in  Fig.  16(a),  can  be  con- 
nected as  in  Figs.  16(6),  (c),  (d)  for  different  types  of  function, 
and  also  in  other  ways  (see  Exercise  4  below). 


(a) 


FiQ.  16. 


Definition.  The  graph  of  an  equation  in  two  variables  is  the 
curve  such  that:  (1)  Every  point  whose  coordinates  satisfy 
the  equation  is  on  the  curve,  and  (2)  Conversely,  the  coordinates 
of  any  point  on  the  curve  satisfy  the  equation. 

To  plot  the  graph  of  an  equation. 

Solve  the  equation  for  one  of  the  variables  in  terms  of  the  other y 
thus  obtaining  one  as  a  function  of  the  other. 

Then  proceed  as  indicated  in  the  rule  for  the  graph  of  a  function. 

In  many  of  the  applications  of  the  methods  of  coordinates, 
the  coordinates  refer  to  quantities  of  different  kinds  such  as 
time,  distance,  work,  cost,  etc.,  and  the  graph  represents  a 
relation  between  two  of  these  quantities. 


EXERCISES 

1.  Construct  the  graphs  of  each  of  the  following  pairs  of  functions  on 
the  same  axes.  State  a  relation  that  each  pair  of  graphs  bear  to  one 
another. 

(a)  3a;,  3x4-2.     (b)  -2a;,-2a;  +  2.     (c)  Kia?-3.     (d)  -'Sx,-3x-2 

(e)  -3x  +  4,  -3x  -  6. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     19 

2.  Construct  the  graphs  of  the  following  pairs  of  functions  on  the  same 
axes.  Can  the  graph  of  one  member  of  a  set  be  moved  so  as  to  coincide 
with  the  graph  of  the  other  member?    If  so,  how  can  this  be  done? 

(a)  2x\  2x^  +  3.  (b)  x^  +  4x  +  4:,  x^  +  4x.  (c)  -  x^,  -x^+4. 

3.  Prove  the  Theorem:  The  graph  of  f{x)  +  k  may  he  obtained  by  moving 
the  graph  off{x)  a  distance  k  in  the  direction  of  the  y-axis. 

4.  Plot  the  graphs  of  each  of  the  functions  in  the  following  sets.  To 
insure  the  proper  connection  of  points  obtained  from  integral  values  of  x 
use  intermediate  values  of  x. 

(a)  x»  -  x^,  2x3  -  3a;2  +  X,  2x»  -  x^  -  x. 

(b)  -  x'  +  x2,  -  2x'  +  3x*  -  X,  -  x^  +  2x»  -  X. 
6.  Plot  the  graphs  of  the  following  equations: 

(a)  X   -  3i/  +  7  =  0.  (b)   y2_  4a;=  4.  (c)  a;^  -  ^/^  =  16. 

(d)  x2  +    y^  =  36.  (e)  y  +  2x2  -  4  =  0.         (f)  ^2  +  9^2  =  36. 

6.  Distribution  of  the  heights  of  12-year-old  boys.  Plot  the  graph  and 
state  one  of  its  characteristics. 

Scinches  }^^'    ^^'    ^^'  ^^'    ^^'    ^^'     ^^'     ^^'     ^'^^  ^^'  ^^'  ^^'    ^^'    ^^• 

S^b^^r   }   ^'     ^'    -^^^  ^^'   ^^'    ^^'  ^^^'  -^^^^  ^^'  ^^'  ^^'  ■^^'     ^'     ^• 

Note.  In  constructing  the  graphs  of  some  functions  it  may  be  desir- 
able to  choose  different  units  of  length  on  the  x-  and  ^-axes. 

7.  The  following  tables  give  the  monthly  receipts  of  eggs  in  the  Chicago 
market  in  1910,  the  price  per  dozen,  and  the  storage  of  eggs  by  months  in 
percentages  of  the  total  annual  storage  by  a  Chicago  firm. 

Month.  J.      F.     M.     A.     M.     J.       J.     A.      S.      O.     N.     D. 

Eggs  in      \ 

thousand    \  72,  140,  160,  760,  500,  400,  300,  240,   180,   120,  80,    48. 

cases.     J 
Storage 
of  eggs 
in  per- 
centages. ^ 

doz'enrce^s.}''^''^'     ^3.     21,    20,      18,     17,     20,    23,     26,30,     33. 

Let  time  be  the  independent  variable  in  each  instance,  and  plot  the 
three  graphs  on  the  same  set  of  axes.  Let  a  convenient  length  from  the 
origin  on  the  ?/-axis  represent  the  three  values,  800  thousand  cases,  100  %, 
and  40  cents  a  doz.^'S,  then  mark  off  the  •  subdivisions  50  thousand  cases, 
5  %,  and  5  cents  a  dozen,  and  multiples  cii  these  subdivisions. 

State  a  relation  that  exists  between  each  two  of  the  three  graphs. 


1.5,    .8,9.5,    42,     19,     22,      1,      0,       1,      .3,    .3,2.6. 


20  ELEMENTARY  FUNCTIONS 

What  inferences  can  be  drawn  from  the  graphs  with  respect  to  the  effect  of 

storing  eggs  on  the  price? 

8.    The  following  are  the  monthly  statistics  for  butter  received,  stored, 

and  the  price  in  the  Chicago  market  in  1910. 
Month.        J.        F.    M.    A.     M.      J.       J.      A.       S.        O.      N.  D. 
Tubs  re- 
ceived in  j-   40,     52,   60,    72,  108,    168,    148,    104,    112,     96,     72,  56. 


thousands. 

p«l}    ''   2-^' 2.8,   3.4,    14,     43,    10,      2,   4.2,   2.5,   4.5,     4. 

Ib^in  cents  }  ^^'     ^^'   ^^'   ^^'    ^^'     ^^'     ^^'      ^^'     ^^'     ^^'     ^^'   ^^' 

Plot  the  three  graphs  on  the  same  set  of  axes,  as  in  problem  7.  State 
a  relation  that  exists  between  each  two  of  the  three  functions.  What  in- 
ferences can  be  drawn  from  the  graphs  regarding  the  effect  of  storage  of 
butter  on  the  price? 

Note.  When  a  law  is  stated  in  such  general  terms  that  numerical  data 
representing  a  concrete  situation  cannot  be  derived,  a  graph  which  will 
picture  the  general  situation  can  be  obtained  by  constructing  a  table  of 
values  with  purely  arbitrary  sets  of  numbers  which  conform  to  the  law. 
Such  a  graph  will  indicate  the  mode  of  change  of  the  function  with  respect 
to  the  independent  variable  without  representing  a  concrete  situation. 

9.  The  Weber-Fechner  law  of  psychology  states  that  as  the  intensity 
of  an  external  stimulus  increases  in  geometric  progression,  the  correspond- 
ing sensation  increases  in  arithmetic  progression.     Construct  the  graph. 

Let  X  represent  the  external  stimuli,  and  y  the  corresponding  sensations. 
Let  the  geometric  progression  1,  2,  4,  8,  ...  be  the  values  assumed  for 
X,  and  the  arithmetic  progression  1,  2,3,  4,  .  .  .  be  the  values  assumed  for  y. 

Plot  the  points  whose  coordinates  are  (1, 1),  (2,  2),  (4,  3),  etc.,  and  draw 
the  graph. 

At  what  value  of  x  should  the  graph  begin?  Can  there  be  an  external 
stimulus  without  a  corresponding  sensation? 

The  law  is  said  to  hold  for  the  senses  of  touch,  hearing  and  seeing,  but 
not  for  taste  and  smell. 

10.  Water  pressure  dies  away  uniformly  because  of  the  resistance  of 
the  conduits.  Construct  a  graph  to  show  the  change  in  water  pressure 
for  points  at  different  distances  from  a  reservoir  situated  on  a  hill. 

11.  Malthus'  law  states  that  population  increases  in  a  geometric  pro- 
gression with  reference  to  time,  while  subsistence  increases  in  arithmetic 
progression.  Plot  and  discuss  the  graphs  with  reference  to  the  influence 
these  two  relations  have  on  one  another.  The  law  of  diminishing  returns 
states  that  after  a  certain  point,  doubling  the  cultivation  in  agriculture  will 
not  double  the  returns.     How  will  this  affect  the  preceding  graphs? 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     21 

Note.  In  the  following  problems  the  functional  relation  changes  in 
character  at  two  points,  and  the  graph  of  the  function  consists  of  several 
distinct  parts. 

12.  The  amount  of  heat  required  to  raise  through  one  degree  the  tem- 
perature of  one  ^ram  of  ice  is  |  a  calorie,  of  one  gram  of  water  is  one  calorie, 
of  one  gram  of  steam  is  |  a  calorie,  approximately.  80  calories  of  heat 
are  absorbed  without  any  rise  of  temperature  when  the  ice  is  melting,  and 
537  calories  without  rise  of  temperature  at  the  boiling  point  when  the 
water  is  vaporizing.  If  the  quantity  of  heat  absorbed  is  regarded  as  a 
function  of  the  temperature  x,  construct  a  graph  representing  roughly 
the  change  from  ice  at  10°  below  freezing  to  steam  at  5°  above  boiling. 
(Centigrade  scale.) 

13.  In  the  case  of  mercury  the  amount  required  to  raise  one  gram  1° 
in  any  of  the  three  forms  is  approximately  .033  calorie,  the  fusing  point 
is  -38°,  the  heat  absorbed  in  fusing  8.8  calories,  the  boiling  point  675°, 
and  the  heat  absorbed  in  vaporization  67.7.  Construct  a  graph  for  mer- 
cury analogous  to  that  for  water. 

14.  A  man  walks  away  from  his  home  at  the  rate  of  4  miles  an  hour  for 
three  hours,  and  then  returns  at  the  rate  of  2  miles  an  hour.  Construct 
a  graph  showing  his  distance  from  home  at  any  time. 

15.  A  man  rides  away  from  a  town  at  the  rate  of  6  miles  an  hour  for 
2  hours.  He  then  stops  for  one  hour,  and  walks  back  at  the  rate  of  3  miles 
an  hour.     Construct  a  graph  showing  his  distance  from  town  at  any  time. 

16.  Construct  on  the  same  axes  the  graphs  of  the  functions  of  x  which 
give  the  perimeter  and  area  of  a  square  whose  side  is  x.  Determine  from 
the  graphs  for  what  values  of  x  the  perimeter  is  (1)  less  than  the  area, 
(2)  equal  to  the  area,  (3)  greater  than  the  area. 

17.  Construct  on  the  same  axes  the  graphs  of  the  functions  which  ex- 
press the  circumference  and  area  of  a  circle  in  terms  of  the  radius.  De- 
termine from  the  graph  for  what  values  of  r  the  circumference  is  (1)  less 
than  the  area,  (2)  equal  to  the  area,  (3)  greater  than  the  area. 

10.  Discussion  of  the  Table  of  Values.  The  considerations 
in  this  section  and  the  section  following  enable  us,  in  many 
cases,  to  abridge  the  labor  of  building  a  table  of  values,  to 
overcome  special  difficulties,  and  to  discover  properties  of  the 
graph.  , 

Example  1.     Construct  a  table  of  values  and  ihe  graph  of 

fix)  =  ix^  -  4. 

Symmetry.  We  shall  first  see  that  the  table  of  values  need  he 
computed  only  for  positive  valiies  of  x. 


22  ELEMENTARY  FUNCTIONS 

Substituting  -x  for  x^  we  have 

/(-x)=i(-a^)^-4  =  iaj2-4=/(x). 

Hence  the  function  has  the  same  value  for  any  two  values 
of  X  which  are  equal  numerically,  but  differ  in  sign,  and  there- 
fore if  {x,  y)  is  a  point  on  the  graph,  so  also  is  (-a;,  y).  These 
points  are  symmetrical  with  respect  to  the  2/-axis,  and  hence 
the  graph  is  also,  in  accordance  with  the 

Definition.  A  curve  is  said  to  be  symmetrical  with  respect 
to  a  line  (or  point)  if  its  points  by  pairs  are  symmetrical  with 
respect  to  that  Hne  (or  point).  The  hne  (or  point)  is  called 
an  axis  (or  center)  of  symmetry. 

Then  if  the  part  of  the  curve  to  the  right  of  the  y-sods  is 
plotted,  the  part  on  the  left  may  be  plotted  by  means  of  the 
symmetry,  and  hence  only  positive  values  of  x  are  needed  in 
the  table.    Now  set 

y=f(x)=ix'-^.  (1) 

Solving  for  x,  

x  =  :^V(2y  +  8).  (2) 

Values  Exclvded.  We  have  agreed  to  neglect  imaginary 
values  of  x  and  y.  If  we  substitute  any  real  value  of  x  in  (1), 
we  obtain  a  real  value  for  y,  and  hence  no  values  of  x  need  be 
excluded.  But  from  (2),  we  see  that  all  values  of  2/  <  -  4,  for 
example  y  =  —  5,  make  x  imaginary.  Hence  if  a  table  of  values  be 
constructed  from  (2)  by  assuming  values  of  2/,  all  values  of  y 
less  than  -  4  must  be  excluded. 

Graphically,  since  no  values  of  x  are  to  be  excluded,  the  curve 
runs  off  indefinitely  to  the  right  and  left.  Since  no  positive 
values  of  y  are  excluded  the  graph  runs  up  indefinitely,  but  as 
values  less  than  —  4  are  excluded,  no  part  of  the  curve  lies  more 
than  4  units  below  the  x-axis. 

Intercepts.  The  coordinates  of  the  points  in  which  a  graph 
cuts  the  axes  are  usually  of  special  significance,  and  they  should 
be  included  in  the  table  of  values. 

For  points  on  the  a::-axis,  2/  =  0,  and  hence  the  abscissas  of 
the  points  where  the  graph  cuts  the  a;-axis  are  obtained  by  set- 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


23 


ting  2/  =  0  in  (2),  which  gives  x  =  ±  Vs  =  ±2.8.  These  ab- 
scissas are  called  the  intercepts  on  the  a;-axis  in  accordance 
with  the 

Definition.  The  intercepts  of  a  curve  on  the  x-axis  are  the 
abscissas  of  the  points  where  the  graph  cuts  the  x-axis,  and 
the  intercepts  on  the  y-axis  are  the  ordinates  of  the  points  of 
intersection  with  the  y-axis. 

Since  x  =  0  ior  all  points  on  the  y-axis,  the  intercepts  on  the 
2/-axis  are  found  by  setting  a;  =  0  in  (1),  which  gives  2/  =  -  4. 

Definition.  A  zero  of  a  function  is  a  value  of  x  for  which 
the  function  is  equal  to  zero. 

Hence  the  zeros  of  f(x)  are  identical  with  the  roots  of  the 
equation  f(x)  =  0.  All  the  zeros  of  a  function  which  are  real 
numbers  are  represented  by  the  intercepts  of  the  graph  on  the 
X-axis. 

We  now  build  the  accompanying  table  and  plot  the  points 
A,  B,  Cf  D,  E,  F,  from  it.    Then  construct  B'  symmetrical  to 
B  with  respect  to  the  y-axia. 
This  is  done  readily  on  cross- 
section  paper  by  counting  the      ^ 
squares  from  B  to  the  y-axjs  — — - 
and  proceeding  an  equal  num-      . 
ber  of  squares  beyond.     Sim-      2 
ilarly,  construct  C,  D',  E',  F'  V8 
symmetrical  to  C,  Z),  E,  F  re-      3 
spectively.      Then    draw    the      ^ 
graph. 

The  reasoning  employed  in 
the  illustration  of  symmetry  is  general,  and  may  be  used  with 
any  function  or  equation.     Hence  the  theorems: 

Theorem  lA.  The  graph  of  Theorem  IB.  The  graph  of 
f{x)  is  symmetrical  mth  respect  an  equation  is  symmetrical  with 
to  the  y-axis  if  respect    to   the   y-axis    if   the 

f{-x)=f{x).  equation   obtained   by   replac- 

ing X  by  -Xj  and  simplifying, 
is  identical  mth  the  given 
equation. 


V 

' 

F' 

\ 

i 

F 

y 
-4 
-3.5 
-2 

0 

0.5 

4 

\ 

/ 

\ 

/ 

) 

^ 

0 

) 

i 

V 

._. 

7 

^' 

-> 

'b 

_ 

u 

Fig.  17. 


24  ELEMENTARY  FUNCTIONS 

The  following  theorems,  whose  proofs  are  left  as  exercises, 
follow  from  the  facts  that  the  points  {-  x,  -  y)  and  (x,  -  y)  are 
symmetrical  to  the  point  (a;,  y)  with  respect  to  the  origin  and 
the  X-axis  respectively. 

Theorem  2A.     The  graph  of        Theorem  2B.    The  graph  of 

fix)  is  symmetrical  with  respect    an  equation  is  symmetrical  with 

to  the  origin  if  respect  to  the  origin  if  the  equa- 

f{—x)  =  —fix),  Hon  obtained  by  replacing  x  by 

—X  and  yby  —y,  and  simplify- 
ing, is  identical  with  the  given 
equation. 
Theorem  SA.     The  graph  of        Theorem  SB.  The  graph  of 
f(x)  is  symmetrical  with  respect    an  equation  is  symmetrical  with 
to  the  x-axis  if  its  values  occur  in    respect  to  the   x-axis    if   the 
pairs  which  are  numerically  equal    equation  obtained  by  replacing 
but  opposite  in  sign.  y  by  —  y,  and  simplifying,  is 

identical  with  the  given  equation. 
We  shall  use  the  phrase  to  discuss  the  table  of  values  of  a  func- 
tion to  mean  that  the 

Symmetry, 

Values  to  be  excluded. 

Intercepts,  and 

Asjnuptotes  (see  next  section) 

are  to  be  determined  before  building  the  table  of  values.  For  the 
last  three  considerations,  solve  the  equation  for  y  in  terms  of  x 
and  for  x  in  terms  of  y.  But  if  the  intercepts  are  desired  in- 
dependently, they  may  be  found  by  setting  either  variable  equal 
to  zero  and  solving  for  the  other. 

EXERCISES 

1.  Does/(-  x)  always  equal  either  ±/(a;)? 

2.  Discuss  the  table  of  values  (omitting  asymptotes)  and  plot  the  graph 
of  each  of  the  functions  and  equations. 


(a)  x^  -  2. 

(b)  xy9. 

(c)  3-x2. 

(d)  y   =  X*. 

(e)  2/'  -  4x  +  2  =  0. 

(f)   2/=x»  +  2. 

(g)  x^  -  9x. 

(h)  2/2  +  6x  =  0. 

(i)   y*-{-4x^0. 

FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


25 


3.    Discuss  the  table  of  values   (omitting  asymptotes)   and  plot  the 
graph  of 


(a)     a:2+    2/2  =  16. 

(b)    x^  +  y^-Qx  =  0. 

(c)  x^  +  y^  +  4x  =  0. 

(d)  ix'-  +    2/2  =  16. 

(e)     a:2_2/2  =  16. 

(f)   a;2  -  1/2  _  6a:  =  0. 

(g)    4X2  ^  Qy2  ^  36. 

(h)  9x2  +  ^2  +  36^  ^  0. 

(i)   y  =x^-Ax\ 

4.  If  J{x)  is  any  one  of  the  functions  whose  graphs  are  given  below, 
determine  whether  or  not  J{- x)  =  */(x),  find  the  value  of  /(O),  and  the 
values  of  x  for  which /(x)  is  zero,  and  for  which  it  is  imaginary. 


Fig.  18. 

11.  Functions  becoming  Infinite.  As3miptotes.  Continuing 
the  discussion  in  the  preceding  section,  in  the  following  ex- 
ample we  shall  need  the 

Definition.  It  is  said  that  a  function  becomes  infinite  as  x 
approaches  a  if  the  numerical  value  of  the  function  can  be 
made  larger  than  any  positive  number,  however  large,  by  giv- 
ing X  a  value  sufficiently  near  to  a. 


26  ELEMENTARY  FUNCTIONS 

Example.  Build  a  table  of  values  and  plot  the  graph  of 
the  function 

v-~  (1) 

Symmetry.    The  tests  for  symmetry  show  that  the  graph  is 
not  symmetrical  with  respect  to  either  axis  or  the  origin. 
Solving  (3)  for  x  we  get 

x  =  ^±^-  (2) 

y  \  J 

Values  excluded.  As  no  radicals  occur  in  (1)  and  (2),  no 
imaginary  values  are  encountered.  Hence  no  values  of  x  or 
y  need  be  excluded  on  this  account,  and  the  graph  runs  indef- 
initely up  and  down,  and  to  the  right  and  left. 

But  the  values  x  =  4  and  t/  =  0  must  be  excluded  as  it  is 
impossible  to  divide  by  zero  (see  3  (c),  page  xv). 

Intercepts.  Setting  a;  =  0  in  (1),  the  intercept  on  the  ?/-axis 
is  y  =  -  h  But  as  2/  =  0  is  an  excluded  value,  the  curve 
does  not  cut  the  x-axis. 

Asymptotes.  To  determine  the  form  of  the  graph  between 
the  points  corresponding  to  a;  =  3  and  x  =  5,  the  table  includes  a 
number  of  pairs  of  values  near  the  excluded  value  a;  =  4. 

0,       1,       2,       3,  3.5,  3.8,     3.9,     4,  4.1,  4.2,  4.5,  5,  6,  7,  8 

4     4'  4'    ^'    "^'  "^'    ~^'    "^^'     "'    ^^'      ^'      ^'  ^'  h  i  I 

The  nimaerical  value  of  the  function,  according  to  the  table, 

increases  as  x  gets  nearer  to  4,  and  it  is  readily  seen  that  by 

giving  x  values  sufficiently  near  4  the  numerical  value  of 

;;  can  be  made  larger  than  any  given  positive  number, 

a;  —  4 

however  large.    Thus  to  make ^  numerically  greater  than 

X  "~  ^ 

10,  let  X  have  any  value  between  3.9  and  4.1  except  the  value 
4;  to  make  it  greater  than  100,  let  x  have  any  value  between 
3.99  and  4.01  except  the  value  4;  etc. 

Hence  the  function -z  ^^^omes  infinite  as  x  approaches  4. 

This  fact  is  indicated  in  the  table  by  the  symbols  (4,  »). 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


27 


In  the  figure  it  is  seen  that  as  the  graph  gets  nearer  the  line 
perpendicular  to  the  x-axis  at  the  point  for  which  a;  =  4,  it 
recedes  farther  and  farther  from  the  x-axis.  Such  a  Hne  is 
called  an  asymptote  in  accordance  with  the 

Definition.  An  asymptote  of  a  curve  is  a  straight  line  such 
that  the  distance  from  a  point  on  the  curve  to  the  line  approaches 
zero  as  the  point  recedes  in- 
definitely from  the  origin  along 
the  curve. 

Similar  reasoning  establishes 
the  fact  that  the  x-axis  is  also 
an  asymptote,  corresponding  to 
the  excluded  value  y  =  0. 

The  example  furnishes  an 
illustration  of  the  general  prin- 
ciple: 

If  a  function  becomes  infinite 
CLS  X  approaches  a,  the  line  per- 
pendicular to  the  X-axis  at  x  =  a 
is  an  asymptote  of  the  graph. 

We  shall  consider  asymptotes 
of  the  graph  of  an  equation  or 
function  only  when  they  are  parallel  to  one  of  the  coordinate  axes. 

Asymptotes  parallel  to  the  y-axis  may  be  found  by  solving  the 
equation  for  y;  the  solution  is  a  function  of  x,  and  to  each  value 
of  X  for  which  this  function  becomes  infinite  there  corresponds  a 
vertical  asymptote.  Horizontal  asymptotes  may  be  found  in  a 
similar  manner  by  solving  the  equation  for  x. 

An  algebraic  function  becomes  infinite  for  real,  finite  values  of 
the  variable  only  if  the  variable  is  contained  in  the  denominator, 
and  if  the  denominator  is  zero  for  one  or  more  real  values  of  the 
variable. 

EXERCISES 

1.   Discuss  the  table  of  values  and  plot  the  graph  of 

1  >  ^  4 


Vi 

■■"T- 

1 
1 

' 

1 
1 

1 

1 

\ 

^ 

1 
1 

V, 

- 

,    . 

o 

] 

I 

^ 

— 

_ 

™i>x" 

— 



... 

1 

. 

\l 

\ 

1 

— 

-€ 

1 

Fig.  19. 


(d)  - 


(b) 


x^  +  1 


(f)  2xy  -  2»  +  3y  »  0. 


28  ELEMENTARY  FUNCTIONS 

(g)  2xy  +  Zx-by  =  0.     (h)  x^  -  2a:y  +  4  =  0.     (i)   a:^  -  4a:?/  +  6  =  0. 

(m)  x^  -^y  +  2x  =  0.        (n)  a;^^/  +  2^  -  3a:  =  0.  (o)  x'y  -  ^y  -  3x  =  0. 

2.  If  a  gas  is  kept  at  the  same  temperature,  the  product  of  the  pressure 
and  the  volume  is  constant.  Discuss  the  table  of  values  and  plot  the 
pressure  as  a  function  of  the  volume,  assuming  a  numerical  value  for  the 
constant. 

12.  Variation  of  a  Function.  Under  this  heading  we  con- 
sider primarily  three  things. 

First.  Sign  of  the  function.  If  x  increases  through  a  given 
interval,  the  sign  of  the  function  will  be  positive  if,  and  only  if, 
the  graph  hes  above  the  a:-axis;  and  the  sign  will  be  negative 
if,  and  only  if,  the  graph  hes  below  the  a:-axis.  For  the  ordinate 
representing  a  value  of  the  function  lies  above  or  below  the 
X-axis  according  as  the  function  has  a  positive  or  negative  value. 

Second.  Changes  of  the  function.  If  x  increases  through  a 
given  interval,  the  value  of  the  function  increases  if,  and  only 
if,  the  graph  rises  as  it  runs  to  the  right;  and  the  value  de- 
creases if,  and  only  if,  the  graph  falls  as»it  runs  to  the  right. 
For  the  ordinate  representing  a  value  of  the  function  increases 
or  decreases  according  as  the  curve  rises  or  falls.  The  motion 
to  the  right  corresponds  to  increasing  values  of  x. 

Third.  Average  rate  of  change  of  the  function.  This  will 
be  considered  in  the  following  section. 

In  order  to  state  in  what  intervals  the  function  is  positive  or 
negative,  or  is  increasing  or  decreasing,  it  is  necessary  to  de- 
termine the  values  of  x  bounding  these  intervals.  These  values 
of  X,  the  corresponding  values  of  the  function,  and  the  points 
on  the  graph  representing  them,  are  the  remaining  important 
elements  of  the  variation  of  the  function.    They  are: 

First:  The  real  zeros  of  the  function,  represented  by  the  in- 
tercepts of  the  graph  on  the  a;-axis. 

Second:  The  values  of  x  for  which  the  function  becomes  in- 
finite, which  give  rise  to  the  vertical  asymptotes. 

Third:  The  maximum  and  minimum  values  of  the  function, 
which  we  proceed  to  define. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     29 

Definition.  A  function  f{x)  is  said  to  have  a  maximum 
value  for  X  =  a  if  /(a)  is  greater  than  all  other  values  of  f(x) 
when  X  is  near  x  =  a.  The  point  on  the  graph  whose  coordi- 
nates are  x  =  a  and  y  =  f(a)  is  called  a  maximum  point 

If  X  increases  through  x  =  a,  the  value  of  f(x)  will  increase  to 
the  value  /(a),  and  then  decrease.  And  the  maximum  point 
will  be  higher  than  the  other  points  on  the  graph  nearby. 

Definition.  A  function  f{x)  is  said  to  have  a  minimum 
valu£  for  X  =  a  if  /(a)  is  less  than  all  other  values  of  f{x)  when 
x  is  near  x  =  a.  The  point  [a,  /(a)]  on  the  graph  is  called  a 
minimum  point. 

If  X  increases  through  x  =  a,  the  value  of  f{x)  will  decrease  to 
the  value /(a),  and  then  increase.  The  minimum  point  will  be 
lower  than  other  points  on  the  graph  nearby. 

Example  1.     Discuss  and  plot  the  graph  of  the  function  2x'^  -  3a;  -  9. 
Let 

y  =  2x2  -  3x  -  9,  (1) 

To  solve  for  x  in  terms  of  y  we  first  write  (1)  in  the  form 

2x^  -Sx  -9  -y  '0. 

Then  by  the  formula  for  solving  a  quadratic 


3  ^V9  +  72  +  8y      3      VSl  +  Sy  ,_. 

^  =  4  =  4  "  4  (2} 

Symmetry.  The  tests  for  symmetry  show  that  the  graph  is  not  sym- 
metrical with  respect  to  either  axis  or  the  origin. 

Valites  excluded.  From  (1),  no  values  of  x  need  be  excluded,  so  that 
the  graph  extends  indefinitely  to  the  right  and  left. 

Equation  (2)  shows  that  we  must  exclude  all  values  of  y  for  which 
81  +  82/  <  0.  Hence  we  must  exclude  y  <  -  10^,  so  that  no  part  of  the 
curve  Hes  more  than  lOi  units  below  the  x-axis. 

Intercepts.  Setting  y  =  0  in  (2),  the  intercepts  on  the  x-axis,  or  the 
zeros  of  the  function,  are  found  to  be 

3  9         .  .       o 

4  4 

Setting  X  =  0  in  (1),  the  intercept  on  the  y-a.xis  is  y  =  -9. 
Asymptotes.     There  are  no  vertical  or  horizontal  asymptotes. 
We  now  proceed  to  build  a  table  of  values  and  draw  the  graph. 
The  variation  of  the  function  is  readily  discussed  in  connection  with 
the  graph. 


30 


ELEMENTARY  FUNCTIONS 


Zeros  of  the  function.    These  are  represented  by  the  intercepts  on  the 
rc-axis,  OA  and  OB.     The  curve  crosses  the  axis  at  A  and  B. 

Sign  of  the  function.  The  graph  is  above  the 
X-axis  to  the  left  of  the  point  A  and  to  the  right 
of  B.    Hence 

2x2  -  3a;  -  9  >  0  if  ^  ^  _  j  ^^    ^^  ^  ^  ^ 

The  graph  is  below  the  x-axis  between  A  and 
B,  and  hence 

2x2  -  3x  -  9  <  0  if  -  1.5  <  X  <  3. 

The  last  set  of  symbols  is  read  "x  is  greater  than 
—  1.5  and  less  than  3." 

Maximum  and  minimum  values.  We  saw  above 
that  all  values  of  y  less  than  -  lOg  must  be  ex- 
cluded. Hence  y  =  -  lOl  is  the  smallest  value  of 
the  functions.  Substituting  this  value  in  (2),  the 
corresponding  value  of  x  is  seen  to  be  f.  Hence 
the  function  has  the  minimum  value  -  10|  when 
X  =  f .  This  value  of  the  function  is  represented 
Fig.  20.  ^y  *^®  ordinate  ED,  and  the  point  Z>(|,  -  10|)  is 

a  minimum  point. 
This  function  does  not  have  a  maximum  value. 

Changes  of  the  function.  Since  the  graph  falls  to  the  right  at  the  left  of 
the  point  D, 

2x2  -  3x  -  9  is  decreasing  as  x  increases  if  x  <  f . 

And  since  the  curve  rises  to  the  right  of  D, 

2x2  -  3x  -  9  is  increasing  as  x  increases  if  x  >  f . 

The  variation  may  also  be  stated  as  follows,  but  care  must  be  exercised 
to  see  that  none  of  the  important  elements  are  omitted. 

For  a  numerically  large  negative  value  of  x  the  function  is  positive. 
As  X  increases  to-  1.5,  /(x)  decreases  to  zero.  It  then  becomes  nega- 
tive, and  continues  to  decrease  until  it  assumes  its  minimum  value  of 
-  lOff  when  x  =  f .  As  x  increases  from  |  to  3,  the  function  is  negative 
and  increases  to  zero.  As  x  increases  beyond  x  =  3,  /(x)  is  positive  and 
increasing. 


y 

\ 

\ 

\ 

\ 

„ 

1 

—1 

*1 

E 

B 

•^ 

r 

0 

1 

3 

.i 

1 

\ 

1 

\ 

\ 

7l 

\ 

'1 

j 

\ 

1 

^ 

<1 

1 

D 

In  this  example,  the  function  does  not  change  sign  unless  x 
increases  througli  a  zero  of  the  function.    But  a  function  may 


f 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS  31 


change  sign  if  x  increases  through  a  value  for  which  the  function 

becomes  infinite.     Thus j,  whose  graph  is  given  on  page  27, 

changes  sign,  from  negative  to  positive,  if  x  increases  through 
the  value  4.  The  intercepts  on  the  a:-axis  and  the  vertical 
asymptotes  should  therefore  be  determined  before  considering 
the  sign  of  a  function. 

In  the  example  just  cited,  the  vertical  asymptote  does  not 
separate  intervals  in  which  the  function  increases  or  decreases, 
but  it  may  do  so.  For  example,  the  function  l/x"^  becomes  in- 
finite if  X  approaches  zero,  and  hence  the  y-axis  is  a  vertical 
asymptote.  As  x  increases,  this  function  increases  if  x  is 
negative,  and  decreases  if  x  is  positive.  This  function  also 
shows  that  a  vertical  asymptote  need  not  separate  intervals  in 
which  the  function  has  opposite  signs,  for  1/x^  is  positive  for 
all  real  values  of  x. 

A  good  order  for  considering  the  elements  of  the  variation  of  a 
function  is  as  follows: 

Zeros  of  the  function, 
Function  becomes  infinite, 
Sign  of  the  function, 
Maxima  and  minima. 
Changes  of  the  function. 

As  the  zeros  and  asymptotes  are  taken  up  in  discussing  the 
ible  of  values,  only  the  last  three  are  new  ideas. 
These  properties  of  a  function  may  be  determined  approxi- 
mately by  inspection  of  the  graph  of  the  function.  This  pro- 
cedure is  especially  useful  if  the  table  of  values  or  the  graph 
be  given  and  the  functional  relation  itself  is  unknown.  The 
accuracy  of  the  results  will  depend  upon  the  choice  of  units  on 
the  axes,  the  care  with  which  the  graph  is  drawn,  and  the 
closeness  with  which  it  is  read. 


32 


ELEMENTARY  FUNCTIONS 


Example  2.  A  thermograph  is  an  instrument  which  records  the  tem- 
perature continuously  by  means  of  such  a  curve  as  in  the  figure.  Discuss 
the  variation  of  the  temperature  as  a  function  of  the  time. 


E 

"1 

id 

/" 

"m 

/ 

\ 

— 

/ 

\ 

-30 

/ 

^ 

^ 

/ 

"^ 

\ 

20- 

/ 

N 

1 

A, 

f 

> 

/ 

-10° 

— > 

I 

1 

/ 

^l 

N 

^ 

aJ 

1 

i 

-0_ 

N 

B 

\>1 

0  A 

.M 

M 

3  A 

\m 

t 

A 

M. 

f/^ 

M. 

S  1 

Too 

% 

31 

jtf 

6P 

M. 

9  P 

M. 

^f'^l 

\ 

!^ 

J 

A 

\ 

ft 

u 

/ 

A 

^ 

— 

— 

— 

X 

1 

Fig.  21. 

Zeros  of  the  function.  The  graph  cuts  the  time  axis  at  B  and  D,  hence 
the  temperature  was  zero  at  3  a.m.  and  at  8:30  a.m. 

Sign  of  the  function.  The  graph  is  above  the  time  axis  from  A  io  B 
and  from  D  to  F  and  below  from  B  to  D.  Hence  the  temperature  was 
above  zero  from  12  Mid.  to  3  a.m.  and  from  8.30  a.m.  to  12  Mid.  the  next 
night,  and  below  zero  from  3  a.m.  to  8:30  a.m. 

Maximum  and  minimum  values.  At  C  the  ordinates  cease  to  decrease 
and  begin  to  increase.  Hence,  the  temperature  had  a  minimum  value 
of  about  -  14°  at  6  a.m.  At  E  the  ordinates  cease  to  increase  and  begin 
to  decrease.  Hence,  the  temperature  had  a  maximum  value  of  about 
+  45"  at  3  p.m. 

Changes  of  the  function.  The  graph  rises  from  C  to  E  and  falls  from 
A  to  C  and  from  E  to  F.  Hence,  the  temperature  increased  from  6  a.m. 
to  3  P.M.  and  decreased  from  12  Mid.  to  6  a.m.  and  from  3  p.m.  to  12  Mid. 

If  it  is  not  desired  to  treat  each  topic  separately,  the  results  might 
be  stated  as  follows: 

The  temperature  at  midnight  was  10°.  It  decreased  until  it  became 
zero  at  3  a.m.  and  continued  to  decrease  until  6  a.m.,  when  it  reached 
a  minimum  value  of  about  -  14°.  It  then  increased,  becoming  zero  at 
8:30  A.M.,  until  3  p.m.,  when  its  maximum  value  was  about  45°.  From 
that  time  on  it  decreased  to  about  20°  at  midnight. 

The  discussion  of  this  function  is  continued  in  the  next  section. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS  33 


EXERCISES 

1.   Discuss  the  table  of  values,  plot  the  graph,  and  determine  the  varia- 
tion of  each  of  the  functions: 


(a) 

y  =x/2. 

(b)  x^. 

(c)  2   -  x\ 

(d) 

y  =  16a:  -  xK 

(e)  xy  +6y  =  4. 

(f)   x2-42/  +  8  =  0. 

(g) 

x"  -2y  +  Ax^Q. 

(h)  2xy   -x  -3y  =  0. 

(i)    x^  -2xy-\-l=^  0. 

(J) 

x^  -5x  -4y  =  0. 

(k)  a;2  -4xy  +  2  =  0. 

(1)   x^  -xy  +  S^O. 

(m) 

xy  =  Q. 

(n)  xh/  =  12. 

(o)  a;  +  2/»  =  0. 

2.  The  surplus  and  shortage  of  railroad  cars  in  thousands  for  the  months 
of  the  years  1915  and  1916  were  as  follows.  Plot  the  graph  and  discuss 
the  function. 

1915.  180,  279,    321,  327, 291, 299,  275,  265,      185,      78,        26,       38. 

1916.  46,    21,  -  20,      3,    33,    57,    52,      9,    -19,    ^60,   -114,  -107. 

3.  The  following  are  the  data  for  the  surplus  reserves  of  New  York 
banks  in  millions  for  the  months  of  the  years  indicated.  Plot  the  graph 
and  discuss  the  function.  The  Federal  Reserve  System  was  inaugurated 
November  16,  1914. 

1914.  29,     32,     20,     17,     38,     40,     15,     -29,    -28,    -0.4,     72,    117. 

1915.  121,  135,  131,  156,  168,  185,  159,     177,     197,      179,  168,    155. 

4.  One  side  of  a  rectangle  whose  perimeter  is  12  inches  is  x.  Find  the 
area  as  a  function  of  x.  Construct  the  graph  of  the  function,  discussing 
the  table  of  values,  and  find  the  value  of  x  if  the  area  is  a  maximum.  What 
is  the  maximum  area? 

5.  A  farmer  wishes  to  fence  off  a  poultry  yard  whose  area  is  to  be  6 
square  rods.  If  one  dimension  is  x,  express  the  perimeter  (the  amount 
of  fencing  needed)  as  a  function  of  x.  Discuss  the  table  of  values  and 
plot  the  graph  of  the  function.  What  will  the  dimensions  be  to  require 
the  least  amount  of  fencing?     How  much  should  he  purchase? 

6.  There  are  a  number  of  diseases  with  continued  fever  in  which  the 
course  of  the  temperature  is  sufficiently  characteristic  to  furnish  the 
diagnosis.     Croupous  pneumonia  is  one  of  these.    (See  Fig.  22.) 

Discuss  the  three  functions.  The  zero  for  temperature  would  be  the 
normal  temperature  98°.4.  By  comparison  of  the  three  functions  state 
in  words  some  of  the  symptoms  of  the  disease. 

7.  The  data  for  the  temperature  curves  of  measles  and  scarlet  fever 


34 


are    given    in    the    following    tables, 
following  graph. 


ELEMENTARY  FUNCTIONS 

Compare    the    graphs   with   the 


J)igeaae 

1 

2 

S 

■f 

5 

e 

7 

8 

9 

10 

11 

12 

70  m 

1 

60  UO 

/ 

sh 

uA 

A 

A 

ir 

y 

V 

V 

/ 

\A 

h\^ 

— 

50  120 

•      r\ 

••— ' 

i^'7 

./^ 

^ 

^m 

"A 

V 

v^ 

./ 

V 

i 

\h 

Jfi  100 

1 
-I 

V^ 

t 

SO    80 

i! 

\\ 

sN 

'7' 

jl 

X^/ 

'rn 

20    60 

i 

V 

-  V— 

103^ 
102" 
lOf 
10(P 
99'* 
98'* 
97'* 


Temperature.,-.^^  BeapiratiotL-.-^ Pulse 

Temperature,  pulse,  and  respiration  tables  in  croupous  pneumonia. 

Fig.  22. 
Day  of  fever.  1  2  3  4  56  7  8 

T.  (Measles)  a.m.      99^.4  101°.2  101°.2  101°.8  103^.4  102'*.8  99°      98^.4 
P.M.     103°.8  ior.8  102^8  104°.8  105^8  103^4  99^8  normal 
T.  (Scarlet  /  98°.4  105^2 

fever)       '    '  \  103°.8  104''.8  104°     103°.6  102°.6  101°.8  100°.8  99°.98 
P.M.     103°.4  105°.8  105°.2  104°.4  103°.4  103°.2  101°.4  99°.4 

Discuss  these  functions.  How  do  the  temperature  curves  distinguish 
pneumonia,  measles,  and  scarlet  fever,  one  from  another? 

13.  Average  Rate  of  Change  of  a  Function.  The  average 
rate  of  change  of  temperature  during  a  given  period  of  time  is  a 
famiUar  idea.  For  instance,  in  Example  2  of  the  preceding 
section,  the  temperature  rose  from  —  14°  to  45°  between  6  a.m. 
and  3  P.M.,  a  total  rise  of  59°  in  9  hours.  Dividing  59  by  9,  we 
see  that  the  average  rate  of  change  in  this  interval  of  time  was 
about  6.5  degrees  per  hour. 

On  the  graph,  C  K  represents  the  change  in  time  from  6  a.m. 
to  3  P.M.,  and  KE  the  corresponding  change  in  temperature. 
Hence  the  average  rate  of  change  of  temperature  in  this  in- 
terval, 6.5  degrees  per  hour,  is  represented  by  the  ratio  KE/C  K. 

The  graph  has  its  most  abrupt  rise  from  about  the  point  G  to 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


35 


Ax 

X 

y 

^y 

^y/^x 

3 

12  Mid. 
3  a.m. 

10 
0 

-10 

-3.3 

9 

6  a.m. 

3  P.M. 

-14 

+45 

+59 

+6.5 

2 

8  a.m. 

10  A.M. 

-6 
+19 

+25 

+12.5 

the  point  H,  which  indicates  that  the  average  rate  of  change  of 
temperature  was  greatest  from  about  8  a.m.  to  10  a.m.  This 
average  rate  of  change  is  represented  by  the  ratio  JH/GJ, 

Let  y  denote  the  temperature 
at  any  time  x,  and  Ai/  the  change 
in  temperature  during  an  inter- 
val of  time  Ao;.  Since  the  aver- 
age rate  of  change  is  the  ratio  of 
the  change  of  temperature  to  the 
corresponding  change  in  time  it 
is  expressed  by  A?//Aa:.  The  computation  of  the  average  rate 
of  change  for  several  different  intervals  may  be  effected  con- 
veniently in  tabular  form,  the  values  of  y  for  the  given  values 
of  X  being  obtained  from  the  graph. 

The  idea  of  the  average  rate  of  change  of  a  function  of  the 
time  has  been  extended  to  include  a  function  of  any  variable. 
This  generalization  is  given  in  the 

Definition.  The  average  rate  of  change  of  a  function  of  Xy 
for  a  particular  change  in  x,  is  the  ratio  of  the  corresponding 
change  in  the  function  to  the  change  in  x. 

If  y  denotes  the  function,  and  Ai/  the  change  in  y  due  to  a 
change  of  Ax  in  x,  then  the  average  rate  of  change  of  y  with  re- 
spect to  X  is  symbolized  by  Ay /Ax. 

If  X  increases  from  one  given  value  to  another,  the  average 
rate  of  change  of  y  may  be  found  as  in  the  illustration  above, 
except  that  the  values  of  y  would  be  computed  from  the  given 
function  instead  of  being  read  from  the  graph.  The  method 
of  finding  a  general  expression  for  the  average  rate  of  change 
for  any  interval  is  illustrated  in  the 


Example.  Find  the  average  rate  of  change  of  the  function  y  =  Ix^  for 
the  intervals  of  x  from  1  to  3,  from  2  to  4,  and  from  2  to  6.  Find  also  the 
average  rate  of  change  for  any  interval. 

The  details  of  the  computation  for  the  three  given  intervals  are  given 
b  the  table  on  page  36. 

To  find  the  average  rate  of  change  of  the  function  for  any  interval,  we 
start  with  any  pair  of  corresponding  values  x  and  y,  and  let  Ay  be  the  change 
in  y  produced  by  a  change  of  Ax  in  a; .     Then  x  +  Ax  and  y  +  Ay  are  cor- 


36 


ELEMENTARY  FUNCTIONS 


responding  values  of  the  independent  variable  and  the  function,  anc 

hence  these  values  satisfy  the  given  equation. 

Therefore  y  +  Ay  =  l{x  +  Ax)^, 

or  y  +  Ay  =  lx^  +  ^x  Ax  +  ^Ax% 

But  y  =  Ix^. 

Subtracting,    Ay  =  ^x  Ax  +  ^Ax^. 

This  gives  the  change  in  y  due  to  a  change 

of  Ax  in  X.     Dividing  by  Ax,  the  requirec 

average  rate  of  change  for  any  interval  Ax  is 

Ax     2""^^^'  ^^' 

The  average  rate  for  any  particular  interval  may  be  obtained  from  thu 
result  by  substitution.    For  example: 

For  the  first  given  interval,  from  1  to  3,  x  starts  with  the  value  x  =  1 
and  increases  by  Ax  =  2.     Substituting  these  values  in  (1),  we  get 
Ay      1.1 


Ax 

x 
1 
3 
2 
4 
2 
6 

y 

1 

21 

1 

4 

1 

9 

Ay 

Ay/Ax 

2 

2 

1 

2 

3 

Ih 

4 

8 

2 

Ax 


2^1  +  4 


X2  =  l. 


For  the  interval  from  2  to  4,  x 
Ay     1 


2,  and  Ax  =  2.    Hence 


Ax     2^2  +  ^x2  =  1-. 


2,  and  Ax  =»  4,  so  that 


^x2  +  |x4  =  2. 


desirable 


And  for  the  interval  from  2  to  6,  x 
Ay 

Ax 
As  these  results  agree  with  those  obtained  above,  we  have 
check  on  the  correctness  of  the  entire  procedure. 

The  graphical  representation  of  the  average  rate  of  change  is  important, 
The  corresponding  pairs  of  values  (x,  y)  and  (x  +  Ax,  y  +  Ay)  are  repre- 
sented by  two  points  P  and  Q  on  the  graph,  so  that 

X  =  OM,  y  =  MP,  x  +  Ax  =  ON,  y  +  Ay  ^  NQ. 

Hence  Ax  =  MN  =  PR,  and  Ay  =  RQ. 

Then  Ay /Ax  is  represented  by 
the  ratio  RQ/PR.  That  is,  the 
average  rate  of  change  of  a  func- 
tion is  represented  graphically  by 
the  ratio  of  the  difference  of  the 
ordinates  of  two  points  on  the  graph 
to  the  difference  of  their  abscissas. 

This  interpretation  holds  for  any 
two  points  P  and  Q  on  the  curve. 
For  no  matter  what  the  relative 
positions  of  the  points  may  be,  we 
have,  by  the  definition  on  page  13, 

OM  +  MN  '  ON        and 


X 

^^ 
-1 

0 
1 
2 
3 
4 
5 
6 


V 

<. 

1 

y 
1 

I 

\ 

f 

1 

1 

1 

/ 

y 

/ 

p 

/ 

R 

N 

y 

1, 

-I 
-J 

J 

0 

1 

ii 

I 

'  ' 

Fio. 
NR  +  RQ^  NQ. 


23. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


37 


EXERCISES 

1.  Find  the  average  rate  of  change  of  each  of  the  functions  below  if 
X  changes  from  0  to  1;  from  1  to  3;  from  1  to  4.  What  can  be  said  of 
these  average  rates?  Find  the  average  rate  for  any  interval,  draw  the 
graph,  and  interpret  the  average  rate  graphically 

(a)  2x-3.  (b)  2x.  (c)    -x    +a. 

(d)  \x  +  2.  (e)     -  ix  +  3.  (f)     -  2a:  +  5. 

2.  Find  the  average  rate  of  change  of  each  of  the  following  functions 
for  the  intervals  from  1  to  3;  from  1  to  -  1;  from  0  to  3;  for  any  interval. 
Check  the  results.     Plot  the  graph  and  interpret  the  average  rates. 

(a)   -  x2  +  16.  (b)  x2  -  2x.  (c)  x2  -  2a;  +  1. 

(d)      a;2  +  2.  (e)  x^  -  5x  +  6.  (f)  a:^. 

*  3.  For  each  of  the  following  functions,  discuss  the  table  of  values,  plot 
the  graph,  and  determine  the  variation,  including  the  average  rate  of  change 
for  any  interval. 

(a)  4  -  x2.  (b)   x2  -  4a;.  (c)    4a;  -  x^. 

(d)  x2  -  4a;  +  4.  (e)    x^  -  3a;  +  2.  (f)     7?  -  3x2. 

4.  Discuss  the  function  represented  by  the  following  temperature 
graph.  Find  the  greatest  and  the  least  average  rate  for  the  three-hour 
intervals  indicated  on  the  time  axis. 


so"                                                                  ^                V 

/                 \ 

-:              \ 

N         I                \ 

\                r                              \ 

i^  -/              \ 

».  ±^.^±               s 

6  PM. 


P.M. 


tl2M.         8  A.M.      6  A.M.       9  A.M.       12  Noon      3  P.M. 
Fig.  24. 
6.  The  daily  variation  of  temperature  for  a  person  in  normal  condition 
18  given  in  the  following  table.     Plot  the  graph  and  discuss  the  function. 
For  what  hour  of  the  day  is  the  average  rate  the  greatest?     The  least? 


Hour                6  A.M. 
Temperature     97''.6 

2 
99° 

7 
98M 

3 
99M 

8 
98°.4 

4 
99°.2 

9         10         11 

98°.7     99°     98°.9 

5           6          7 

99°.3     99°.2     99° 

12  m.   1p.m. 
98°.8     98°.9 

8           9 
98°.8     98°.7 

10 

98°.4 

11 

98°.2 

12 

98° 

1  A.M.       2 

97°.6     97°.6. 

The  temperature  from  2  a.m.  to  6  a.m.  is  stationary. 


fe 


38 


ELEMENTARY  FUNCTIONS 


14.  Classification  of  Functions.  In  discussing  functions  it 
is  convenient  to  separate  them  into  classes  according  to  the 
properties  they  possess  or  the  character  of  the  operations  that 
are  involved  in  calculating  them. 

The  following  scheme  indicates  the  important  divisions  and 
subdivisions  of  the  functions  which  we  shall  study  in  this  course. 

Linear 


1.  Algebraic 
Functions 


Rational 


Irrational 


Integral 


Quadratic 
Cubic 

Biquadratic 
Polynomial 


Fractional 


Elementary 


Exponential 
Logarithmic 
2.  Transcen-  "^      Trigonometric 

dental  I  Inverse  Trigonometric 

Functions     Other  transcendental  functions  studied  in  higher  mathe- 
^  matics. 

An  algebraic  function  *  is  a  function  whose  value  may  be 
computed  when  that  of  x  is  given  by  the  application,  a  finite 
number  of  times,  of  the  operations  of  algebra,  namely,  addi- 
tion, subtraction,  multiplication,  division,  involution,  and 
evolution. 

The  following  are  examples  of  algebraic  functions: 

2x  +  3,     ax'^  +  bx  +  c,     Vx^  -  9,     ?^^- 

X  —  fj 

In  the  work  of  this  course  only  real  numbers  will  be  used. 
Hence,  for  us,  such  an  algebraic  function  as  Vl6  —  x^  is  defined 
only  for  values  of  x  such  that  -4  =  a;  =  +  4  (read  **  x  is  greater 

*  This  definition  is  sufficiently  general  for  the  purposes  of  this  course. 
The  definition  used  in  higher  mathematics,  which  is  more  inclusive  is 
as  follows:  Given  a  polynomial  in  y, 

fiy)  =  (W  +  flilT"*  +  fW^  +    .  .  .  +  an-\y  +  an, 

whose  coefficients  oo,  ai,  02,  •  •  •  On-i,  fln  are  polynomials  in  x,  tnen  any 
solution  of  the  equation  f{y)  =  0  for  2/  in  terms  of  a;  is  called  an  algebraic 
function.    . 


ffi 


FUNCTIONS,  EQUATIONS/AND  GRAPHS  39 

than  or  equal  to  -  4,  and  less  than  or  equal  to  +  4  ")>  since 
for  all  other  values,  as  a;  =  5,  the  value  of  the  function  is  im- 
aginary. 

An  integral  rational  function  (integral  function  or  polynomial) 
is  a  function  whose  value,  for  a  given  value  of  x,  may  be  found 
by  the  operations  of  addition,  subtraction,  and  multipUcation 
applied  a  finite  number  of  times.     The  following  is  the  general 

form: 

2/  =  ax"  +  6x"-i  -\-  . . .  +  kx  -\- 1 

Polynomials  are  classified  according  to  the  degree  of  the 
highest  power  of  x  occurring. 

Linear  function  ax  +  b. 

Quadratic  function  ax^  -\-bx  +  c. 

Cubic  function  ox^  +  bx"^  -\-  ex  -{■  d. 

Biquadratic  function  ax*  +  6x^  4-  cx^  -\-  dx  +  e. 

A  rational  function  is  a  function  whose  value,  for  a  given  value 
of  x,  may  be  found  by  the  four  rational  operations  of  addition, 
subtraction,  multiplication,  and  division,  appUed  a  finite  num- 
ber of  times.  If  division  involving  x  in  the  divisor  occurs  in 
the  computation,  the  function  may  be  expressed  as  the  quotient 
of  two  polynomials,  and  it  is  then  called  a  fractional  function. 
The  general  form  is 

'^  ^  ba^  +  6ix"»-i  +  . . .  +  bm-ix  +  bm 

An  irrational  function  is  a  function  which  involves  tne  extrac- 
tion of  roots  in  addition  to  the  four  rational  operations. 

In  contrast  with  algebraic  functions,  all  other  functions  are 
called  transcendental.  This  term  is  merely  a  synonym  for 
non-algebraic. 

Inverse  functions.  If  y  be  given  as  a  function  of  x  by  the 
relation 

2/  =  x2  -  1, 


X  is  also  a  function  of  y  obtained  by  solving  this  equation  for  z, 
namely, 

X  =  =t  Vy  -h  1. 


40  ELEMENTARY  FUNCTIONS 


The  two  functions  x^  -  1  and  =«=  Vy  +  1  are  called  inverse 
functions. 

The  independent  variable  in  the  first  is,  as  usual,  x,  but  in 
the  second  it  is  y.  In  order  to  write  the  inverse  function  with 
X  as  the  independent  variable  we  must  replace  y  by  x.  Hence 
the  inverse  of  x^  -  1  is  =*=  Va;  +  1. 

Definition.  If  y  be  set  equal  tof(x),  the  equation  solved  for 
X  in  terms  of  y,  and  y  replaced  by  x  in  the  final  result,  then  the 
function  of  x  so  obtained  is  called  the  inverse  oi  f{x). 

The  same  result  is  obtained  by  interchanging  x  and  y  in  the 
equation  y  =  /(x),  and  then  solving  for  y  in  terms  of  x. 


EXERCISES 

1.   What  kind  of  a  function  is 

(a)  3x  +  2?  (b)      x^  -4x  +  1?  (c)   a^  -2x^+  7? 


4x  -2 


(d)^^?  (e)   Vx»-4? 

2.  Give  a  numerical  example  of  a 

(a)  quadratic  function.  (b)  fractional  function, 

(c)  cubic  function.  (d)  irrational  function. 

3.  Find  the  inverse  of  each  of  the  following  functions;  classify  eacb 
function  and  its  inverse. 

(a)  x3  +  3.        (b)  Vx2  -  4.     (c)  \/^^-        (d)  vT+x+ VFT^. 

(e)  y  =-  y/x  +  Vx. 

4.  Find  the  inverse  functions  defined  by  the  following  equations  and 
classify  them: 

(a)    xy  +  2y  -X  -3  =  0.  (b)   x^ +  2xy  +  3y^  +  I  =  0. 

(c)  xy"  "k.  (d)  x^  +  y^  =  a'. 

15.  Summary.  Suppose  that  the  data  of  a  law  of  a  science 
can  be  recorded  in  a  table  of  values  of  two  variables,  or,  by  some 
mechanical  device,  in  the  form  of  a  curve.  The  generalization 
which  expresses  the  law  connecting  the  two  variables  is  a  func- 


I 


Ip 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     41 


tional  relation  (Sections  1-4).  If  this  relation  can  be  expressed 
in  mathematical  symbols,  then  mathematics  becomes,  to  a 
considerable  extent,  the  language  of  that  science.  An  important 
problem,  therefore,  is  the  determination  of  an  expression  in 
mathematical  symbols  for  a  function  which  is  given  otherwise 
(Section  6;  see  also  5  and  6  below). 

Function  —  table  of  values  —  graph.  These  are  merely  dif- 
ferent manifestations  of  the  same  concept.  A  functional  re- 
lation expresses  a  law  connecting  each  pair  of  numbers  of  two 
sets;  the  table  gives  particular  pairs  of  these  numbers;  and 
the  graph  affords  a  geometric  representation  of  them. 

The  fundamental  problems  arising  in  connection  with  this 
concept  are  the  determinations  of  any  two  of  these  manifesta- 
tions from  the  third.     They  are  six  in  number. 

1.  Given  a  function,  to  build  the  table  of  values.  This  is 
an  easy  process  for  simple  algebraic  functions. 

2.  Given  a  table  of  values,  to  plot  the  graph.  This  is  also 
easily  done  if  the  table  is  sufficiently  extensive. 

3.  Given  a  function,  to  draw  the  graph.  This  is  done  by 
means  of  1  and  2  (Section  9). 

4.  Given  a  graph,  to  construct  a  table  of  values.  Pairs  of 
values  may  be  read,  approximately,  directly  from  the  graph. 

5  and  6.  Given  a  table  of  values,  to  find  the  function.  This 
is  precisely  the  problem  which  confronts  the  scientist  in  seeking 
an  unknown  law,  and  it  is  by  far  the  most  difficult  of  these 
problems.  Methods  of  solution  will  be  considered  in  the  follow- 
ing chapters  (see  Sections  25,  44,  and  90). 

In  Sections  10-13  we  have  considered  properties  of  a  function, 
its  table,  and  its  graph,  which  are  important  in  the  study  of 
any  function.  The  correspondence  between  these  properties  of 
a  function  and  its  graph  may  be  exhibited  compactly  as  follows : 

PROPERTY  OF  GRAPH  PROPERTY  OF  FUNCTION 

Length  of  an  ordinate.  A  value  of  the  function. 

Graph  symmetrical  respect  y-axis.       J{- x)  =  j{x). 
Graph  symmetrical  respect  origin.       f{-x)  =  -  f(x). 
Vertical  lines  do  not  cut  graph.  Excluded    values    of   x.      Function 

imaginary  or  infinite. 


42 


ELEMENTARY  FUNCTIONS 


Intercepts  on  the  a:-axis. 
Intercepts  on  the  y-axis. 
Vertical  asymptote. 
Horizontal  asymptote. 

Graph  above  a;-axis.  - 
Graph  below  x-axis. 
Ordinate  of  maximum  point. 
Ordinate  of  minimum  point. 
Graph  rises  to  the  right. 
Graph  falls  to  the  right. 
Ratio  of  difference  of  ordinates 
to  difference  of  abscissas. 


Zeros  of  function,  i.e.,  j{x)  =  0. 
Values  of /(O). 
Function  becomes  infinite. 
Value  of  function  as  x  becomes  in- 
finite. 
Function  is  positive,  i.e.,  /(x)  >0. 
Function  is  negative,  i.e.,  j{x)  <0. 
Maximum  value  of  function. 
Minimum  value  of  function. 
/(x)increases  as  x  increases. 
/(x)  decreases  as  x  increases. 
Value  of  Ay /Ax.     Average  rate  of 
change  of  function. 


Interpretation  of  a  graph.  The  properties  of  the  graph  may 
be  determined  by  inspection,  if  only  one  has  in  mind  what  to 
look  for,  and  the  corresponding  properties  of  the  function  may 
then  be  stated.  It  is  true  that  the  properties  of  the  graph  are 
proved  by  first  establishing  the  properties  of  the  function. 
But  after  the  graph  is  drawn,  this  interpretation  of  the  graph 
affords  a  comprehensive  point  of  view  of  many  properties  of 
the  function  and  furnishes  a  simple  means  of  stating  any  par- 
ticular property. 

As  we  come  to  study  any  one  class  of  functions  defined  in 
Section  14,  we  shall  take  up  the  properties  listed  above,  and  in 
addition  the  characteristic  properties  which  distinguish  that 
class  of  functions  from  others.  A  typical  graph  for  each  class 
of  functions  should  he  fixed  in  mind,  as  it  enables  one  to  tie  to- 
gether and  recall  quickly  the  characteristics  of  a  function  as 
soon  as  it  is  classified.  This  is  of  great  importance  in  analyz- 
ing problems. 

From  time  to  time  we  shall  add  other  properties  of  graphs  and 
functions  to  the  list  above. 

We  shall  also  take  up  the  relations  between  certain  pairs  of 
functions  and  their  graphs,  which  enable  us  to  obtain  the  graph 
of  one  function  from  that  of  the  other.  Thus  the  graph  of 
f{x)  +  k  may  be  obtained  from  that  of  f(x)  (see  Exercise  3, 
page  19).  These  two  sets  of  relations  constitute  the  framework 
of  the  entire  course. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS 


43 


We  turn  next  to  the  detailed  study  of  particular  classes  of 
functions,  beginning  with  the  linear  function,  which  occurs 
frequently  in  the  applications  of  mathematics. 


MISCELLANEOUS  EXERCISES 

1.   State  as  many  properties  as  possible  of  the  functions  whose  graphs 
are  given  below.  * 


1             y. 

\                            (a) 

I    ^ 

^^»^:^    -* 

^7  {    V  ^ 

\ 

^ 

\ 

T     5^     -i 

±           it      'Ml 

-X     '      t 

-  V^^V  '  ^ 

\^y «  Ky    ^ 

V- 

1     / 

t              '^(c) 

t       t 

t       t        _^ 

1"!^ 

7    1 

_r  1 

/     I 

V 

(d) 

0 

/ 

S 

•"^ 

^^ 

s 

/ 

X 

__^ 

T  '  T 

4^     X 

/i       /I  (.) 

^^  .^fc 

-^XoZX   ^ 

i   /  '  i   /^' 

-U     ^ 

^     ^ 

X     it 

Jl      IL 

"          ^ 

(c)       r 

7 

f 

_.Z 

0     .       2c 

y 

■ 

<« 

/ 

\ 

0 

r 

^ 

V 

y 

. 

\ 

/ 

X 

V' 

(f) 

/ 

\ 

^ 

-_ 

0 

1 

X 

u 

(i) 

—1 

^ 

^ 

' 

/ 

0 

/ 

*x 

/ 

Fig.  25. 

2.  Discuss  the  table  of  values,  plot  the  graph,  and  determine  the  vari- 
ation, including  the  average  rate  of  change,  of: 

(a)  |x  -  3.  (b)  x2  -  42/  -  3  =  0.  (c)  x^  -  16x  +  3y  =  0. 

3.  Discuss  the  table  of  values,  plot  the  graph,  and  determine  the  varia- 
tion, omitting  the  average  rate  of  change,  of: 

(a)  2/2  +  2x  -  9  =  0.  (h)  x^  +  y^  -4x  =  12.  (c)  x^  +  ^y^  -  ix  =  16. 

(d)  x2  -  2x2/  +  5  =  0.  (e)  x"  +  4c2/  +  2  -  0.    (f )  3x2/  +  4x  -  22/  -  0. 


I 


44 


ELEMENTARY  FUNCTIONS 


(g)  y^  =  x«. 
a)  2/^(4 -x) 


(h)  x^  -4y  -1 

x2(4  +  x).  (k)  x3  +  &r  -  16 


■  0.    (i)  x^y  -  42/  -  X  =  0. 
0.    (1)   xy^  +  x  -4y  =  0. 


4.  For  a  given  abscissa,  the  ordinate  of  a  point  on  the  graph  of 
y  =f(x)  +  g{x)  is  the  sum  of  the  ordinates  of  the  points  with  the  same 
abscissa  on  the  graphs  of  /{x)  and  g{x).  Hence 
the  graph  of  y  may  be  obtained  by  drawing  the 
graphs  of  f{x)  and  g{x)  and  then  adding  the  ordi- 
nates of  points  with  the  same  abscissa.  Using  this 
method,  which  is  called  the  addition  of  ordinates, 
construct  the  graphs  of  the  following  functions. 

(a)  y  =  x^  +  ^x.  Solution.  Draw  the  graphs  of 
x2  and  of  \x.  If  MP  and  MQ  are  the  ordinates  of 
points  on  these  graphs  with  the  same  abscissa  OM^ 
then  MR  =  MP  +  MQ  =  MP+  PR  is  the  ordinate 
of  a  point  on  the  graph  of  the  given  function. 

Several  points  on  the  graph  may  be  obtained  in 
this  way,  and  a  smooth  curve  drawn  through  them. 
Notice  that  if  OM  is  negative,  so  also  is  MQ,  so  that  MR<MP. 


r 

y 

T 

V 

/ 

\\ 

"y 

h 

V 

LP 

/ 

1 

\ 

\ 

/; 

r 

R 

N^ 

/ 

/ 
f 

\ 

s 

// 

\ 

y 

M 

K; 

^^. 

^ 

/-'' 

M 

-• 

^^ 

^ 

X 

U 

^ 

Fig.  26. 


(b)  x2  +  X. 

(e)  xV3  +  x/3. 


(c)  a;V3  +  2x, 
(f)    lA  +  x/2. 


(d)  xy2  +  X. 
(g)  a:2  -  2x. 


5.  Express  the  area  of  a  rectangle  as  a  function  of  one  of  its  sides,  as- 
suming that  the  perimeter  is  8  feet.  Plot  the  graph  and  discuss  the  varia- 
tion of  the  function. 

6.  A  house  stands  50  feet  from  the  street.  A  man  is  walking  along  the 
street.  Express  his  distance  from  the  house  as  a  function  of  his  distance 
from  the  entrance  to  the  grounds,  which  is  directly  in  front  of  the  house. 
Discuss  and  plot  the  graph  of  the  function. 

7.  A  man  6  feet  tall  walks  away  frpm  a  lamp-post  12  feet  high.  Ex- 
press the  length  of  his  shadow  as  a  function  of  his  distance  from  the  post, 
and  plot  the  graph  of  the  function. 

8.  In  Exercise  7,  express  the  distance  from  the  post  to  the  shadow  of 
the  man's  head  as  a  function  of  the  man's  distance  from  the  post.  Plot 
the  graph,  and  show  that  the  average  rate  of  change  of  the  function  is 
constant. 

9.  A  point  lies  at  a  distance  r  from  the  origin.  Find  the  equation  ex- 
pressing the  functional  relation  between  the  coordinates  x  and  y  of  the 
point. 

10.  A  man  walks  for  2  hours  at  the  rate  of  3  miles  an  hour,  stops  ai 
hour  and  a  half  for  lunch,  and  walks  back  at  the  rate  of  2  miles  an  houri 
Construct  a  graph  showing  his  distance  from  the  starting  point  at  an] 
time. 


FUNCTIONS,  EQUATIONS,  AND  GRAPHS     45 

Note.  —  A  solution  of  two  simultaneous  equations  in  x  and  y 
consists  of  a  pair  of  values  of  these  variables  which  satisfy- 
both  equations.  If  the  numbers  are  real,  the  point  whose 
coordinates  are  such  a  pair  of  values  is  on  both  graphs.  There- 
fore a  real  solution  of  two  simultaneous  equations  in  x  and 
y  is  represented  by  a  point  of  intersection  of  the  graphs.  To 
find  the  coordinates  of  the  points  of  intersection  of  two  graphs, 
solve  their  equations  simultaneously. 

11.  Plot  the  graphs  of  the  following  equations  and  find  the  coordinates 
of  their  points  of  intersection. 

(a)  a;  +  2y-3  =  0,    x  +  2?/ -  8  =  0. 

(b)  2x  +  y  =  0,      y^  =  4x. 

(c)  a;2  +  2/2  =  25,     3x  +  4y  =  0, 

(d)  x^  +  y^  =  16,    2x-Sy  =  4. 

(e)  x2  +  y2  ^  9,    y^  =  4x. 

12.  Plot  the  graphs  of  the  following  equations  and  from  them  read 
approximate  values  of  the  solutions  of  the  equations. 

4r 


(a) 

0:2  +  42/2  =  16,     2/' =  -^3 

(b) 

xy-y^  =  Q,    y^^x'. 

(c) 

y^a^-Ax^    2x-y  +  2  =  0. 

(d) 

a^  +  2/  =  7,       y^  +  x  =  n. 

13.  Two  engines  in  a  freight  yard  are  running  on  parallel  tracks.  Their 
distances  from  a  signal  tower  (in  hundred  yards)  at  the  time  t  (in  minutes) 
are  given  respectively  by  the  equations 

s  =  _  if  +  4        and        s  =  -  it^  +  4t  +  4:. 

Plot  the  graphs,  using  i  =  0,  4,  8,  etc.     Determine  when  and  where  the 
engines  are  beside  each  other  (two  solutions) . 


CHAPTER  II 


LINEAR  FUNCTIONS 


16.  Uniform  Rate  of  Change.  The  position  of  a  body  mov- 
ing on  a  hne,  straight  or  curved,  is  commonly  indicated  by  its 
distance  s  from  a  given  point  on  the  Hne.  The  body  is  said  to 
move  uniformly,  or  at  a  constant  rate,  along  the  hne  if  the  ratio 
of  any  change  in  s  to  the  corresponding  change  in  time  is  con- 
stant (that  is,  if  this  ratio  has  the  same  value  for  all  intervals 
of  time).  If  As  is  the  change  in  s  during  the  time  A^,  then  for 
uniform  motion  As/ At  =  v,  a  constant  called  the  velocity.  The 
value  of  V  gives  the  change  in  s  during  a  unit  of  time. 

Since  As/ At  is  the  average  rate  of  change  of  s  during  the 
interval  of  time  At  (see  Section  13),  the  above  definition  is 
equivalent  to  the  statement  that  a  body  moves  uniformly  at 
the  rate  of  v  units  of  distance  per  unit  of  time  if  the  average 
rate  of  change  of  s  is  constant  and  always  equal  to  v. 

Example.     A  farmer  lives  10  miles  from  town.     He  starts  from  home 
and  drives  away  from  town  at  the  uniform  rate  of  5  miles  an  hour.     Con- 
struct and  interpret  a  graph  show- 
l   ing  his  distance  from  town  at  any 
time. 

Let  s  denote  his  distance  from 
town  t  hours  after  starting.  Then 
8  —  10  denotes  the  distance  he 
travels  in  t  hours,  and  hence 

s-10 


=  5, 


whence 


s  =  5t  +  10. 


(1) 


In  plotting  the  graph  of  equation 
(1)  we  take  values  of  t  as  abscissas  ac- 
cording to  a  well-established  custom.    The  pairs  of  values  of  t  and  s  in  the 
table  are  represented  by  the  points  A,  B,  C,  D,  E,  which  appear  to  lie  on 

46 


LINEAR  FUNCTIONS 


47 


a  straight  line,  I.  We  assume  that  the  graph  is  indeed  straight,  an  as- 
sumption which  will  be  justified  in  Section  20. 

The  interpretation  of  the  coordinates  of  any  point  on  the  graph  is  that 
the  ordinate  represents  the  man's  distance  from  town  at  the  time  repre- 
sented by  the  abscissa. 

The  interpretation  of  the  velocity  is  worthy  of  detailed  consideration. 
The  successive  values  of  At  given  in  the  table  are  represented  by  AF 

=  BG  =  CH  =  DI  =  I,   and   those   of    As  by   FB  =  GC  =  HD  =  IE  =  5. 

Hence  v  =  As/ At  is  represented  by  any  one  of  the  ratios 

FB      GC  _HD     IE 
AF^  BG~  CH~  Dl' 

Now  consider  any  interval  of  time  beginning  at  d  and  ending  at  ^2. 
The  distances  from  town  at  these  times  are  respectively,  from  (1), 


and 

S2 

=  5^1  +  10, 
=  5t2  +  10. 

Subtracting, 

As  = 

S2  - 

si  =  5(t,  - 

ii) 

whence, 

V 

-  Af  "  ^• 

5At, 


Since  AC  =  At  and  CB  =  As,  v  is 
represented  by  the  ratio  CB/AC. 

We  may  therefore  say  that 
the  velocity  is  represented  by  the 
ratio  of  the  difference  of  the  ordi- 
nates  of  any  two  points  on  the 
line  to  the  difference  of  the  ab- 
scissas. 

Many  other  examples  of  uni- 
form change  with  respect  to 
time  will  readily  suggest  them- 
selves. Thus  we  speak  of  the 
rate  at  which  a  tank  is  filled 
with  water,  the  rate  at  which 
product,  etc. 


Fig.  28. 


At 

t 

8 

As 

As 

At 

0 

10 

1 

1 

15 

b 

b 

1 

?, 

20 

b 

b 

1 

8 

25 

b 

b 

1 

4 

30 

b 

5 

a  manufacturer  turns  out  his 


A  generalization  of  uniform  rate  of  change  with  respect  to  time 
is  given  in  the 


48  ELEMENTARY  FUNCTIONS 

Definition.  It  is  said  that  any  variable  y  changes  uniformly 
with  respect  to  a  second  variable  x,  of  which  y  is  a  function,  if 
the  ratio  of  any  change  in  y  to  the  change  in  x  producing  it, 
Ay /Ax,  is  equal  to  a  constant,  which  is  called  the  rate  of  change 
of  y  with  respect  to  x.  This  rate  gives  the  change  in  y  due  to  a 
unit  change  in  a;. 

For  example,  if  mercury  is  poured  into  a  vertical  tube,  the 
weight  of  the  mercury  in  the  tube  is  a  function  of  the  height 
of  the  column  of  mercury.  The  ratio  of  any  change  in  the 
weight  to  the  change  in  height  is  constant,  and  equal  to  the 
weight  of  a  column  of  unit  height.  Hence  the  weight  changes 
uniformly  with  respect  to  the  height. 

It  follows  from  the  definition  that  if  y  changes  uniformly  with 
respect  to  x,  equal  changes  in  x  produce  equal  changes  in  y. 

EXERCISES 

It  is  assumed  that  the  graphs  in  the  following  exercises  are  straight  lines. 

1.  Solve  the  example  in  Section  16  if  the  man  starts  toward  town, 
considering  the  motion  for  4  hours. 

2.  A  through  freight  train  was  90  miles  from  a  city  at  2  o'clock,  and 
150  miles  at  4  o'clock.  Construct  the  graph  of  its  motion.  Find  its 
velocity,  and  interpret  it  geometrically.  From  the  graph  find  how  far 
the  train  was  from  the  city  at  noon,  and  when  it  passed  through  the  city. 

3.  A  tank  full  of  water  is  being  emptied.  After  5  minutes  it  contains 
150  gallons,  and  after  12  minutes  108  gallons.  Construct  a  graph  show- 
ing the  amoimt  of  water  in  the  tank  at  any  time.  Find  the  capacity  of 
the  tank,  the  time  required  to  empty  it,  and  the  rate  at  which  it  is 
being  emptied.     What   represents  each  of  these  quantities  graphically? 

4.  A  man  walks  up  a  hill  inclined  at  30°  to  the  horizon.  Find  the  rate 
at  which  his  altitude  increases  with  respect  to  the  distance  he  walks.  Is 
it  essential  that  he  walk  with  a  constant  velocity? 

5.  A  coal  wagon  is  being  filled  with  coal.  Would  the  rate  of  change 
of  the  weight  of  the  coal  in  the  wagon  with  respect  to  the  volume  of  the 
coal  be  constant?     If  so,  how  would  it  be  measured? 

6.  Mention  some  quantity  which  is  changing  uniformly  with  respect 
to  some  other  quantity,  different  from  time,  and  state  what  the  rate  of 
change  would  be. 

7.  On  the  same  axes,  plot  graphs  showing  the  number  of  minute  spaces 
the  hands  of  a  clock  pass  over  in  t  minutes,  starting  at  noon  and  ending 
at  1  o'clock.  Determine  from  them  how  many  spaces  the  hands  are 
apart  at  12:30. 


LINEAR  FUNCTIONS 


49 


^ 


17.   Characteristic  Property  of  a  Straight  Line.    Let  Pi  and 

Pi  be  any  two  points  on  a  straight  line,  MiPi  and  M2P2  their 
ordinates,  and  let  Pi  Q  be  perpendicular  to  M2P2.  Then 
Ax  =  PiQ  is  the  difference  of  the  abscissas  of  Pi  and  P2,  and 
A?/  =  QP2  is  the  difference  of 
the  ordinates.  The  ratio  of  y 
the  difference  of  the  ordinates 
to  that  of  the  abscissas  is 

A^^QPa 

^x    PiQ 

For    any   second    pair    of 
oints  on  the  line,  P3  and  P4, 
the  value  of  this  ratio  is 

Ay 
Ax 


Fig.  29. 


PP4 
P3R 


Since  the  triangles  P1QP2  and  PzRPa  are  similar  (why?)> 
we  have 

PP4  ^  QP2 

.PzR      PiQ' 
and  hence 

The  ratio  of  the  difference  of  the  ordinates  of  any  two  points 
on  a  straight  line  to  the  difference  of  the  abscissas,  Ay /Ax,  is  the 
same  no  matter  what  two  points  on  the  line  he  chosen. 

This  fact  has  been  illustrated  in  the  preceding  section. 

Conversely,  a  line  is  a  straight  line  if  the  ratio  of  Ay /Ax  is 
always  the  same  for  any  two  points  on  the  line. 

Let  Pi  and  P2  be  two  definite  points  and  P3  any  third  point 
on  the  given  line  which  is  to  be  proved  straight.  That  the  given 
line  is  straight  will  be  proved  by  showing  that  P3  hes  on  the 
straight  hne  determined  by  Pi  and  P2.  Draw  the  hne  through 
Pi  parallel  to  the  x-axis  cutting  the  ordinates  of  P2  and  P3  at 
Q  and  R  respectively.  Since,  by  hypothesis,  the  values  of 
Ay /Ax  computed  for  Pi  and  P2  and  for  Pi  and  P3  are  equal,  it 
follows  that 

QP2  _  PP3 
PiQ     PiR' 


y' 

^^^"^3- 

-"t^i'i 

1          1 
1          1 

^ 

Ml             4      Ms 

» 

50  ELEMENTARY  FUNCTIONS 

Hence  the  triangles  P1QP2  and  PiRPz  are  similar  (why?), 
and  therefore  Z  QP1P2  =  Z  RP1P3.    Then  P1P3  coincides  with 

P1P2,  and  hence  P3  hes  on 
the  straight  hne  P1P2. 

Definition.  The  con- 
stant value  of  the  ratio  of 
the  difference  of  the  ordi- 
nates  of  any  two  points  on 
a  straight  Une  to  the  differ- 
ence of  the  abscissas  is  called 
Fig  30.  the  slope  of  the  line. 

If  the  slope  is  denoted  by 
m,  this  definition  may  be  expressed  in  the  symbolic  form 

Ay 

For  any  graph,  the  ratio  of  the  difference  of  the  ordinates 
to  the  difference  of  the  abscissas,  Ay /Ax,  represents  the  average 
rate  of  change  of  a  function.  Hence  the  facts  proved  above 
may  be  stated  as  the 

Theorem.  If  y  is  a  function  of  x,  the  graph  of  y  is  a  straight 
line  if,  and  only  if,  the  average  rate  of  change  of  y  with  respect 
to  X  is  constant. 

To  plot  the  graph  of  a  function  we  usually  express  the  func- 
tional relation  in  the  form  of  an  equation,  build  the  table  of 
values,  and  plot  the  curve.  This  process  is  unnecessary  if  it 
Ls  known  that  the  average  rate  of  change  of  the  function  is 
constant,  for  the  theorem  just  proved  shows  that  the  graph  is 
a  straight  line.  To  draw  the  graph  it  is  sufficient  to  obtain 
two  pairs  of  values,  plot  the  points  representing  them,  and 
draw  the  straight  line  through  these  points. 

Example.  Commercial  alcohol,  95%  pure,  is  poured  into  a  bottle. 
Construct  a  graph  showing  the  amount  of  alcohol  in  the  bottle  as  a  func- 
tion of  the  amount  of  liquid  in  the  bottle.     Interpret  the  slope. 

Plot  the  amount  of  liquid,  I,  on  the  horizontal  axis,  and  the  amount  of 
alcohol,  a,  on  the  vertical  axis.  If  Ao  is  the  amount  of  alcohol  in  Ai  units 
of  the  liquid,  we  have  An  =  0.95Ai,  since  95  %  of  the  liquid  is  alcohol. 
Hence  Ao/Ai  =  0.95,  so  that  the  rate  of  change  of  a  with  respect  to  I  is 


LINEAR  FUNCTIONS 


51 


constant.    The  graph  is  therefore  a  straight  line.     If  there  is  no  liquid  in 

the  bottle,  there  is  no  alcohol,  and  hence  the  origin  (0,0)  is  on  the  line; 

and  if  the  bottle   contains   10  units  of 

liquid,  there  are  9.5  units  of  alcohol  in 

the  bottle.     Hence   (10,  9.5)  is  a  second 

point  on  the  line,  and  these  two  points 

are  sufficient  to  determine  the  graph. 

The  slope  represents  the  rate  of  change 
of  a  with  respect  to  I,  Aa/Al  =  0.95,  which 
gives  the  percentage  of  alcohol  in  the  liquid. 

In  plotting  the  graph  by  using  the  point 
(10,  9.5),  it  is  assumed  that  the  unit  of 
volume  is  some  such  unit  as  an  ounce  or 
cubic  centimeter.  If  a  bottleful  were 
chosen  as  the  unit,  which  is  convenient  in 

many  problems,  it  would  be  better  to  use  the  point  (1, 0.95)  and  choose  the 
unit  on  the  Z-axis  as  large  as  convenient.  For  then  I  =  1  would  mean  that 
the  bottle  was  full. 

18.  Slope  of  a  Straight  Line.  "  Slope  of  a  line  "  (defini- 
tion, Section  17)  is  the  term  used  technically  by  mathematicians 
for  what  might  be  called  the  measure  of  steepness  of  the  hne. 
It  may  represent  many  things.    Thus  in  the  example  in  Section 

2/f 


r 

~^ 

(10,^^.5), 

/ 

c 

/ 

/ 

/ 

y^ 

/ 

—5 

A 

/ 

—3 

/ 

/ 

/ 

/ 

i , 

O 

_ 

.,    1 

_ 

— 

\  t'?'! 

Fig.  31. 


y 

F 

^ 

^y^ 

Ay 

Pr 

^ 

^ 

Q 

^ 

Ax 

0 

^ 

fj 

ik 

U    ^ 

Fig.  32. 

16,  As/Ai  is  the  slope  of  the  Hne,  and  hence  the  velocity  v  is 
represented  by  the  slope. 

If  the  slope  m  is  computed  from  two  points  Pi  (xi,  yi)  and 
^2(0:2,  2/2)  on  the  hne,  we  have,  in  either  figure, 

m  =  ^  -  ^^^  -  ^^^^  ~  ^^^^  -  y^-y^  _  yi  -^2 

Ax      PiQ      OM2  -  OMi      X2  -  xi      xi  -  X2 
In  finding  Ay  and  Ax  it  is  essential  that  both  coordinates  of  Pi  be  sub- 
tracted from  those  of  P2,  or  vice  versa. 


52 


ELEMENTARY  FUNCTIONS 


Theorem  1.  A  line  runs  up  to  the  right  or  dovm  to  the  right 
according  as  its  slope  is  positive  or  negative. 

For  two  points  Pi  and  P2  may  be  chosen  on  the  hne  so  that 
Pi  lies  to  the  left  of  P2,  whence  Aa;  is  positive.  Then  Ay  will 
be  positive  or  negative  according  as  P2  is  above  or  below  Pi. 
Hence  m  =  Ay /Ax  will  be  positive  or  negative  according  as 
P2  is  above  or  below  Pi,  that  is,  according  as  the  hne  runs  up 
to  the  right  or  down  to  the  right. 

The  precise  relation  between  the  value  of  m  and  the  direc- 
tion of  the  hne  as  determined  by  the  angle  the  hne  makes  with 
the  X-axis  involves  a  transcendental  function,  and  will  be  con- 
sidered in  a  later  chapter  (see  also  Exercises  3  and  4  below). 

Theorem  2.  If  two  lines 
are  parallel  their  slopes  are 
equal,  and  conversely. 

The  triangles  P1QP2  and 
Pi'Q'P^  are  similar,  since 
their  sides  are  parallel  by 
pairs.    Hence 

QP2_qPl 
PiQ     Pi'Q" 
that  is,  the  slopes  of  the 
two  Hnes  are  equal. 
The  proof  of  the  converse  is  left  as  an  exercise. 
Construction.     To  construct  a  line  through  a  given  point  Pi 
whose  slope  is  a  given  positive  fraction  a/b,  take  Q  b  units  to  the 
right  of  Pi,  and  P2  a  units  above  Q;  then  P1P2  is  the  required  line. 
Yov  we  have 

eP2     a 


Fig.  33. 


m  = 


PiQ'h 


If  the  slope  is  negative,  take  P2  below  Q.  If  the  slope  is  an 
integer,  it  may  be  regarded  as  a  fraction  with  unit  denominator. 

If  two  points  are  close  together,  the  hne  through  them  can- 
not be  drawn  as  accurately  as  if  they  were  farther  apart.  Hence, 
in  this  construction,  it  is  sometimes  desirable  to  take  PiQ  equal 
to  some  multiple  of  b  and  QP2  equal  to  the  same  multiple  of  a. 


LINEAR  FUNCTIONS  53 

EXERCISES 

1.  Plot  the  following  pairs  of  points  and  the  lines  determined  by  them. 
Find  Ax,  Ay,  and  the  slope  m,  and  indicate  the  graphical  significance  of 
each  of  these  quantities. 

(a)  (1,  2)  and  (4,  5).     (b)  (1,  2)  and  (4,  1).         (c)  (1,  2)  and  (-  2,  0). 
(d)  (1,  2)  and  (3,  4).     (e)   (3,  -  2)  and  (5,  2).     (f)   (2,  5)  and  (  4,  2). 

2.  Construct  the  lines  through  the  points  indicated  with  the  given  slope. 

(a)  P(2,  3),  m  ==  i       (b)  P(3,  -  1),  m  =  -  ^       (c)  P(0,  4),  m  =  -  |. 
(d)  P(3,  0),  m  =  2.       (e)   P(-  2,  4),  m  =  f .  (f)   P(5,  2),  m  =  0. 

3.  Show  that  of  two  lines  through  the  same  point  the  one  which  makes 
the  greater  acute  angle  with  the  a;-axis  has  the  larger  slope  numerically. 

4.  Show  that  the  slope  of  a  line  parallel  to  the  x-axis  is  zero. 

6.  If  a  telegram  containing  10  words  or  less  can  be  sent  to  a  certain 
place  for  S5^,  and  a  24-word  message  for  63ff,  construct  a  graph  showing 
the  cost  of  a  message  containing  any  number  of  words,  and  determine  the 
charge  for  each  additional  word  above  10.     What  represents  this  charge? 

6.  Construct  a  graph  to  show  the  cost  of  any  number  of  eggs  at  40^ 
a  dozen.  Determine  from  the  graph  the  number  of  eggs  which  can  be 
purchased  for  70)!f.    What  does  the  slope  represent? 

7.  Construct  a  graph  to  show  the  amount  of  silver  in  any  amount  of  an 
alloy  containing  25%  of  silver.  Determine  from  the  graph  how  much 
silver  there  is  in  20  pounds  of  the  alloy.     What  does  the  slope  represent? 

8.  A  pound  of  an  alloy  contains  3  parts  of  silver  and  5  of  copper.    Con- 
:  struct  a  graph  to  show  the  amount  of  silver  in  any  amount  of  the  alloy. 

Determine  from  the  graph  how  much  of  the  alloy  contains  4 1  pounds  of 
i  silver,  and  how  much  silver  there  is  in  10  pounds  of  the  alloy.  Interpret 
j  the  slope. 

j  9.  Ten  ounces  of  alcohol  95  %  pure  are  poured  into  a  bottle  and  then 
I  5  ounces  of  water  are  added.  Construct  a  graph  showing  the  amount  of 
I  alcohol  in  the  bottle  during  the  process  as  a  function  of  the  amount  of 
I  liquid. 

I  10.  Solve  Exercise  9  if  5  ounces  of  50  %  alcohol  are  added  instead  of  5 
I  ounces  of  water. 

I  11.  Is  the  slope  of  the  line  joining  (3,  6)  to  (6, 1^)  the  same  as  the  slope  of 
[the  line  joining  (3,  6)  to  (  -  3,  -6)?  What  can  be  said  of  the  three 
I  points? 

12.  Are  the  points  (6,  1),  (-  2,  -  3),  and  (0,  -  2)  on  a  straight  line? 

13.  Show  that  the  points  (2,  1),  (3,  7),  (5,  3),  and  (0,  5)  are  the  vertices 
of  a  parallelogram. 

14.  Find  the  value  of  b  if  the  line  determined  by  (2,  h)  and  (-  1,  3)  ia 
parallel  to  the  line  joining  (-4,  -  5)  and  (7,  -  2). 


54 


ELEMENTARY  FUNCTIONS 


15.  Two  boys  start  from  the  same  point  and  run  in  the  same  direction, 
one  at  the  rate  of  15  feet  per  second,  the  other  at  the  rate  of  20  feet  per 
second.  Construct  the  graphs  showing  the  distances  from  the  starting 
point  to  each  boy  at  any  time.  What  do  the  slopes  represent?  De- 
termine from  the  graphs  how  far  apart  the  boys  are  after  3  seconds.  Solve 
the  same  problem  if  the  boys  run  in  opposite  directions. 

16.  Water  is  admitted  into  a  tank  through  two  pipes  at  the  rates  of 
3  and  5  gallons  per  second.  On  the  same  axes,  plot  graphs  showing  the 
amount  which  has  entered  through  each  pipe  at  any  time,  and  also  the 
total  amount.     What  do  the  slopes  represent? 

17.  If  the  cost  of  setting  type  for  a  circular  is  50  cents,  and  if  the  cost 
of  paper  and  printing  is  half  a  cent  a  copy,  construct  a  graph  showing  the 
cost  of  any  number  of  copies.  From  it  determine  the  cost  of  500  copies. 
What  does  the  slope  represent?  Hint:  The  cost  of  setting  the  t3rpe  may 
be  regarded  as  the  cost  of  zero  copies. 

18.  If  $100  is  deposited  in  a  bank  to  draw  simple  interest  at  6  %,  con- 
struct a  graph  to  show  the  amount  at  any  time.  What  does  the  slope 
represent?  The  intercept  on  the  "  amount  axis  "  ?  Determine  graphically 
how  long  it  would  take  the  principal  to  double  itself.  What  is  the  relation 
between  the  amount  and  the  time? 

19.  One  of  the  horizontal  boards  in  a  flight  of  stairs  is  called  a  tread, 
and  one  of  the  vertical  boards  a  riser.  How  can  the  steepness  of  the 
stairs  be  expressed  in  terms  of  the  widths  of  the  treads  and  risers?  j 

20.  Show  that  the  average  rate  of  change  of  a  function  other  than  aj 
linear  function  is  represented  by  the  slope  of  a  secant  Une  of  the  graph,     j 

19.  Graphical  Solution  of  Problems  Involving  Functions 
which  Change  Uniformly.    Graphical  methods  are  used  freely 

by  engineers  and  others  because 
of  their  relative  simplicity.  A 
graphical  solution  of  a  problem  is 
also  often  used  as  a  simple  means 
of  checking  the  accuracy  of  some 
other  solution.  A  method  of  ob- 
taining approximate  solutions  of 
many  problems  by  means  of 
graphs  is  illustrated  in  the  foUo^j 
ing  problems: 

Example  1.    A  druggist  has  a  50' 
solution  of  a  certain  disinfectant  an^ 
also  a  10  %  solution.     How  much  of  the  former  must  be  added  to  3  pint 
of  the  latter  to  obtain  a  20  %  solution? 


d 

1 

/" 

';.. 

L 

V  7 

/_ 

A\ 

/ 

/I 

0  i 

/ 

y 

^  / 

/ 

/ 

g 

/    _^ 

/ 

eL^^ 

"^ 

/  y 

^ 

^' 

/  ^- 

^ 

D           H 

1 

3          — ' 

T 

Fig.  34. 


LINEAR  FUNCTIONS 


55 


The  graph  representing  the  amount  of  disinfectant,  d,  in  a  quantity, 
I,  of  the  50  %  solution  is  the  straight  line  OA  through  the  origin  whose 
slope  is  0.5.  Similarly,  the  graphs  for  10%  and  20%  solutions  are  re- 
spectively OB  and  OC  whose  slopes  are  0.1  and  0.2. 

On  the  Z-axis  take  OD  =  3,  and  draw  the  ordinate  DE  to  the  line  OB. 
Then  DE  represents  the  amount  of  disinfectant  in  3  pints  of  the  10% 
solution. 

Through  E  draw  EF  parallel  to  OA.  Then  the  ordinate  of  a  point  on 
EF  represents  the  amount  of  disinfectant  in  the  mixture  as  some  of  the 
50  %  solution  is  added  to  the  3  pints  of  the  10  %  solution. 

If  DE  cuts  OC  at  G,  and  if  GH  be  drawn  perpendicular  to  the  Z-axis, 
then  in  OH  pints  of  the  combined  mixture  there  are  HG  pints  of  disinfec- 
tant; and  since  G  lies  on  OC,  the  strength  of  the  combined  mixture  is  20  %. 

Hence  we  must  add  DH,  or  1  pint  approximately,  of  the  50  %  solution. 

Example  2.  A  freight  train  goes  from  ^4  to  £  at  the  rate  of  15  miles 
per  hour.  Four  hours  after  it  starts,  an  express  train  leaves  A  and  moves 
at  the  rate  of  45  miles  per  hour. 
The  express  arrives  at  B  half 
an  hour  ahead  of  the  freight. 
How  far  is  it  from  A  to  B  and 
how  long  does  it  take  each  train 
to  go? 

Let  abscissas  denote  time  in 
hours  measured  from  the  time 
of  departure  of  the  freight 
train,  and  ordinates  distances 
from  A. 

The  graph  representing  the 
distance  of  the  freight  at  any 
time   is   the  straight    line  OC 

through  the  origin  with  the  slope  15,  and  that  of  the  express  is  the  line 
DE  through  the  point  D(4,  0)  with  the  slope  45. 

At  B  the  express  is  half  an  hour  ahead  of  the  freight,  and  hence  we  must 
determine  a  point  on  OC  half  a  unit  to  the  right  of  DE.  To  do  this,  take 
F  on  the  <-axis  so  that  DF  =  |.  Draw  the  line  through  F  parallel  to  DE, 
and  let  it  meet  OC  at  G.  If  the  line  through  G  parallel  to  the  Z-axis  cuts 
DE  at  H,  then  HG  =  |.  Hence,  the  required  distance  AB  is  represented 
by  the  ordinates  IH  =  JG  =  100  miles  approximately. 

The  time  of  the  freight  is  represented  by  OJ  =  6.7  hours,  approximately, 
and  that  of  the  express  by  DI  =  2.2  hours,  approximately. 

The  line  FG  may  be  interpreted  as  the  graph  of  a  train  leaving  half  an 
hour  later  than  the  express  and  moving  at  the  same  rate  of  45  miles  an 
hour.  By  the  conditions  of  the  problem,  such  a  train  would  overtake  the 
freight  at  B. 


^^                              Wf¥^ 

Z                               W 

-^0                                -^d.  L 

y^Li 

-60                                                           X             /      / 

-.^                tt~~~ 

-^            tt 

-«.     ^/            ri    "" 

-  ^            Tl     "" 

^                           .£Lji          1      1 

"      ^^±_l-J_±Zl 

Fig.  35. 


56  ELEMENTARY  FUNCTIONS 


EXERCISES 


Solve  each  of  the  following  exercises  graphically.  Use  a  sharp,  hard 
pencil,  and  choose  the  units  on  each  axis  so  that  the  figure  will  be  as  large 
as  convenient. 

1.  Two  trains  start  from  the  same  place  at  the  same  time  and  run  in 
the  same  direction  at  the  rates  of  30  and  50  miles  per  hour  respectively. 
How  far  apart  will  they  be  at  the  end  of  two  hours?  When  will  they  be 
100  miles  apart? 

2.  Solve  Exercise  2  if  the  trains  run  in  opposite  directions. 

3.  For  printing  business  cards  one  firm  charges  75^5  for  setting  the  type, 
and  a  apiece  for  the  cards  and  printing.  Another  firm  makes  no  charge 
for  typesetting,  but  charges  f  ^  apiece.  Which  firm  will  print  more  cheaply 
100  cards?  500  cards?  Which  will  do  the  more  printing  for  $1.25?  For 
$2.00?  How  many  cards  will  both  print  at  the  same  cost,  and  what  will 
the  cost  be? 

4.  A  man  rides  horseback  at  the  rate  of  8  miles  an  hour  for  an  hour 
and  a  half,  and  then  walks  back  at  the  rate  of  4  miles  an  hour.  Another 
man  starts  from  the  same  place  at  the  same  time  and  walks  in  the  same 
direction  at  the  rate  of  3  miles  an  hour.  When  and  where  will  they 
meet? 

5.  How  much  water  must  be  added  to  10  gallons  of  alcohol,  95  %  pure, 
to  reduce  it  to  a  66|  %  solution? 

6.  A  farmer  sells  three  and  a  half  dozen  eggs  to  a  grocer  at  25}*  a  dozen, 
and  takes  his  pay  in  rice  at  8ff  a  poimd.  How  many  pounds  of  rice  will 
he  receive? 

7.  At  what  time  are  the  hands  of  a  clock  together  between  three  and 
four  o'clock?  When  are  they  opposite?  Hint:  The  position  of  either 
hand  may  be  determined  by  the  number  of  minute  spaces  it  has  passed 
over  since  leaving  "12."  Plot  this  number  as  a  function  of  the  number 
of  minutes  after  three  o'clock. 

8.  Solve  Exercise  7  for  eight  and  nine  o'clock. 

9.  The  earth  revolves  around  the  sun  in  52  weeks,  and  the  planet 
Mars  in  98  weeks.  How  often  is  the  earth  between  Mars  and  the 
sun? 

20.  Graph  of  the  Linear  Function  mx  +  b.  Let  the  function 
be  denoted  by  2/,  so  that 

y  =  mx  +  b,  (1) ' 

Let  us  first  determine  the  average  rate  of  change  of  y  with 
respect  to  x.    Let  the  independent  variable  change  from  an^ 
value  re  to  X  +  Arc,  and  suppose  that  the  dependent  variable 


LINEAR  FUNCTIONS  57 

changes  from  y  to  y  +  Ay.    Then  the  corresponding  values 

X  +  Ax  and  y  +  Ay  satisfy  (1),  so  that 

y  +  Ay  =  m{x  +  Ax),  (2) 

Subtracting  (1)  from  (2), 

Ay  =  m  Ax, 

and  dividing  by  Ax, 

Ay /Ax  =  m. 

Hence  the  average  rate  of  change  of  y  with  respect  to  x  is 
constant,   and  the  graph  of  (1)   is  therefore  a  straight  Une 
:  (Theorem,  Section  17). 

I      If  the  graph  is  a  straight  Hne  the  value  of  Ay /Ax  is  the  slope 

I  of  the  Une  (Definition,  Section  17),  and  hence  the  slope  of  the 

I  graph  of  (1)  is  m.    The  constant  term  b  is  the  intercept  on 

the  ?/-axis,  for  if  a;  =  0,  then  from  (1)  y  =  b.    The  hne  is  thus 

determined  by  the  point  (0,  6)  and  the  slope  m  (Construction, 

Section  18). 

We  may  therefore  express  the  fundamental  theorem  of  this 
chapter  as  follows: 

Theorem  1.     The  graph  of  the  linear  function  mx  -{•  b,  or  of 
the  equation  y  =  mx  +  b,  is  the  straight  line  whose  slope  is  m, 
i  and  whose  intercept  on  the  y-axis  is  b. 

If  6  =  0,   the  hne  passes  through  the  origin.     Hence  the 
important 

Corollary  1.     The  graph  of  the  function  mx,  or  of  the  equation 
y  =  mx,  is  the  straight  line  through  the  origin  whose  slope  is  m. 
Corollary  2.     The  graphs  of  the  equxitions 

y  =  mx  +  b        and        y  =  mx  +  b' 

are  parallel  lines. 

An  equation  of  the  first  degree  in  x  and  y  is  called  linear, 
and  may  be  reduced  to  the  form 

Ax  -h  By  +  C  =  0.  (3) 

Theorem  2.     The  graph  of  any  linear  equation  (3)  is  a  straight 
lin£. 


58  ELEMENTARY  FUNCTIONS 

If  the  equation  contains  y,  so  that  B  9^  0  (read  "  B  is  not 
equal  to  zero")>  we  may  divide  by  B  and  solve  for  y,  obtaining 
the  linear  function  ^        q 

whose  graph  is  the  straight  Hne  whose  slope  is  m  =  —  A/B, 
and  whose  intercept  on  the  y-axis  is  6  =  -  C/B  (Theorem  1). 
li  B  =  0,  equation  (3)  has  the  form  Ax  +  C  =  0,  whence,  if 

^  ^  ^^  a;  =  -  CM. 

The  graph  of  this  equation  is  a  straight  line  parallel  to  the 
?/-axis.  For  all  points  whose  abscissas  have  the  same  value, 
-C/A,  no  matter  what  the  ordinates  may  be,  are  equidistant 
from  the  ^/-axis. 

Hence  the  graph  of  (3)  is  always  a  straight  line. 

Corollary  1.  The  graph  of  an  equation  of  the  form  x  ^  a  con- 
stant is  a  straight  line  parallel  to  the  y-axis,  and  that  of  y  =  a 
constant  is  a  line  parallel  to  the  x-axis. 

The  proof  of  the  first  statement  appears  explicitly  in  the 
preceding  proof.  Is  that  of  the  second  included  in  the  proof  | 
(see  Exercise  4,  Section  18)?  | 

Corollary  2.  //  a  linear  equation  be  solved  for  y,  then  the] 
coefficient  of  x  is  the  slope  of  the  graph  of  the  equation,  and  the 
constant  term  is  the  intercept  on  the  y-axis. 

In  constructing  the  graph  of  a  linear  equation  it  is  necessary  to 
plot  hut  two  points.  The  intercepts  (Section  10)  usually  fur- 
nish the  two  most  convenient  points.  The  line  may  be  drawn  j 
more  accurately  by  choosing  points  some  distance  apart  thanj 
by  taking  them  close  together. 

The  graph  of  y  =  mx  may  he  drawn  by  plotting  but  one  point 
and  joining  it  to  the  origin. 

EXERCISES 

1.   Construct  the  graphs  of  the  following  equations: 
(a)  2/  =  2x  +  5.  (h)    y=-2x  +  5.  (c)    yx  +  Z. 

id)  y=  -x  +  7.  (e)     y  =  3a;.  (f)     ?/  =  2a;  -  4. 

(g)  a;  -  2j/  +  6  -  0.         (h)    2a;  +  Sy  -  12  =  0.  (i)  2x  +  3y  +  12  = 

G)   1/  =  |ar.  (k)     y  =  ix  +  2.  (1)     z/  =  -  ^a;  +  5. 


LINEAR  FUNCTIONS 


59 


2.  In  experiments  on  the  temperature  at  various  depths  in  a  mine,  the 
temperature  t  (in  degrees  Centigrade)  was  found  to  be  connected  with  the 
depth  d  (in  feet)  by  the  equation  ^  =  20  +  0.01  d.  Construct  the  graph. 
What  is  represented  by  the  ordinate  and  abscissa  of  a  point  on  the  graph? 
by  the  intercept  on  the  f-axis?  by  the  slope? 

3.  In  experiments  on  a  pulley  block  the  pull  P,  in  pounds,  required  to 
lift  a  weight  of  W  pounds  was  found  to  be  P  =  0.3  W  +  0.5.  Construct 
the  graph.  What  does  the  slope  represent?  The  intercept  on  the  P-axis? 
What  is  the  value  of  the  rate  of  increase  of  P  with  respect  to  Wl 

4.  The  readings,  2\  of  a  gas  meter  being  tested,  were  found  in  com- 
Darison  with  those  of  a  standard  meter,  S,  and  the  two  sets  of  readings 
satisfied  the  equation  T  =  300  +  1.2  S.  Draw  the  graph.  What  do  the 
ibscissa  and  ordinate  of  a  point  on  it  represent?  The  slope?  The  in- 
■ercept  on  the  7'-axi3?  Has  the  consumer  whose  meter  is  being  tested 
)een  getting  more  or  less  gas  than  the  meter  indicated? 

I    5.  Plot  the  graph  of  the  equation  s  =  -3^  +  12.     Describe  the  motion 

I  or  the  first  6  hours  if  s  represents  the  distance  of  a  man  from  town,  in 

i  niles,  at  the  time  t,  in  hours.     Interpret  the  slope  and  the  intercepts  on 

i  )oth  axes. 

i    6.  The  ordinate  of  a  point  on  one  of  the  graphs  in  the  figure  represents 

s  he  distance  s  (in  feet)  of  a  moving  body  from  a  certain  station  at  the  time 

i     (in    seconds)     repre- 

jented  by  the  abscissa. 

!  )escribe  the  motion 
epresented  by  each  of 

jhe  graphs  (a),  (6),  (c), 

I  d),    from    t  =  -Q    to 

;  -12. 

I  Solution  for  (a).  From 
he  graph,  s  =  7  when 
«  -  6.    Hence  the  body 

Uarts  at  a  point  7  feet 

i*om  the  station  on  the 

I  ositive  side  of  it.    Ast 

I  icreases,    s    decreases, 

j  nee  the  graph  is  falling 
3  it  runs  to  the  right 

jhanges  of  the  function).  Hence  the  body  is  approaching  the  station. 
Then  t  =  0,  the  distance  from  the  station  is  s  =  4  (interpretation  of  the 
itercept  on  the  s-axis).  The  body  passes  through  the  station  when 
«  8,  for  then  s  =  0  (interpretation  of  the  intercept  on  the  i-axis).  Four 
jconds  later,  that  is,  when  t  =  12,  the  body  has  reached  a  point  2  feet 
syond  the  station  in  the  negative  direction;  for  when  t  =  12,  s  =  -  2. 
The  rate  of  change  of  s  with  respect  to  t,  namely,  the  velocity,  is  con- 


8^ 

V 

^ 

c^ 

^ 

^ 

"V 

K 

^ 

^ 

U) 

X 

^ 

2 

< 

*^ 

^ 

^-' 

^ 

N 

.^ 

^ 

' 

> 

^ 

<^ 

t*.. 

(. 

k. 

K 

^. 

^ 

-> 

0 

.-^ 

"^ 

/ 

s 

0 

'i 

-i 

"•^ 

X 

■^^ 

\ 

. 

(d) 

^ 

^ 

-^ 

^ 

^ 

(c) 

_ 

_J 

Fig.  36. 


60 


ELEMENTARY  FUNCTIONS 


Btant  since  the  graph  is  a  straight  line, 
points  (0,  4)  and  (8,  0),  we  get 

_  4f      0-4 

"~  At~  S-0' 


Computing  its  value  from  tl 


Solving  the  equation  for  TF,  we  get  TT  =  ^ D  +  t- 


The  negative  sign  indicates  that  the  total  change  in  s  in  any  interval 
of  time  A^  is  negative,  and  hence  the  body  is  moving  in  the  negative 
direction  at  the  uniform  rate  of  half  a  foot  per  second. 

7.  In  the  following  equations,  s  denotes  the  number  of  feet,  measured 
along  the  road,  at  the  time  t  (in  seconds),  from  a  tree  to  a  steam  roller. 
Construct  the  graph  of  each  equation,  and  from  it  describe  the  motion  of 
the  steam  roller  from  f  =  -  5,  to  i  =  10. 

(a)  s  =  2t  +  S.  {h)  s=  -^t  +  S.  (c)  s  =  -  2<  -  4. 

8.  Construct  and  interpret  the  graph  of  each  of  the  equations  below,  in 
which  W  denotes  the  part  of  a  piece  of  work  that  a  man  can  do  in  D  days. 

(a)  20W  -  D  -  5  =  0. 

^      "      Hence  the  graph  is 

the  straight  line  whose  slope  is  ^V  and  whose  intercept  on  the  TF-axis  is  \. 

Since  W  =  1  indicates  that  the  piece  of  work  has  been  completed,  the 

graph  need  be  plotted  only 
from  W  =  OtoW  =  1. 

The  intercept  on  the 
TF-axis  indicates  that  one 
fourth  of  the  job  has  been 
completed  when  we  begin 
to  consider  the  problem, 
and  the  slope  represente 
the  rate  at  which  the  man 
works,  namely,  ^V  o^  the 
piece  of  work  per  day. 

When  TF  =  1,  the  graph 
shows  that  D  =  15  (which 

may  be  verified  from  the  equation),  that  is,  it  will  take  the  man  15  days  tc 

complete  the  work. 

If  we  care  to  interpret  negative  values  of  D,  the  graph  shows  that  wher 

TF  =  0,  D  =  -  5.     Hence  if  we  regard  the  whole  piece  of  work  as  performec 

by  the  man  in  question,  he  started  5  days  ago. 

(b)  12TF  -  D  -  6  =  0.  (c)  lOTF  -  3D  =  0. 

(d)  lOTF  -  2D  +  4  =  0.  (e)  30TF  -  5D  +  10  =  0. 

9.  Mention  some  function  of  the  form  mx  arising  in  daily  Ufe,  and  die 
cuse  it  graphically. 


Fig.  37. 


fl 


LINEAR  FUNCTIONS  61 

10.  Mention  some  function  of  the  form  mx  +  b  arising  in  daily  life, 
and  discuss  it  graphically. 

11.  Plot  the  graphs  of  the  equations  below,  using  a  single  set  of  co- 
ordinate axes  for  each  case.  Each  figure  will  contain  several  lines  cor- 
responding to  the  values  of  the  constants  indicated. 

(a)  y  =  mx  for        w  =  |,  |,  1,  2,  4. 

(h)  y  =  2x  +  b         for         5  =  -  3,  -  2,  0,  2,  5. 

(c)  y  =  mx  +  2        for        m  =  |,  1,  3,  -  3,  -  1,  -  J. 

12.  Exhibit  in  tabular  form  as  many  as  possible  of  the  corresponding 
properties  of  a  straight  line  and  the  function  fix)  =  mx  +  b  oi  which  the 
line  is  the  graph  (see  Section  15).  Indicate  special  properties  of  this 
function  which  distinguish  it  from  all  other  functions. 

21.  Variation.  We  have  already  had  a  number  of  illustra- 
tions of  the  important  relation  y  =  mx,  in  which  one  variable 
is  a  constant  times  the  other.  For  example,  the  price  paid 
for  eggs  is  the  price  per  dozen  times  the  number  of  dozen  bought; 
if  a  man  walks  at  a  uniform  rate,  the  distance  is  the  rate  times 
the  time;  etc.    Two  technical  terms  are  used  in  this  connection. 

Definition.  A  variable  is  said  to  vary  as,  or  to  be  propor- 
tional to,  a  second  variable  if  the  first  is  equal  to  a  constant 
times  the  second.  SymboHcally,  y  varies  as  x,  or  y  is  propor- 
tional to  X,  ii  y  =  mx,  m  being  a  constant. 

This  use  of  the  word  proportional  is  justified  by  the  fact  that 
•  two  values  of  y  are  proportional  to  the  corresponding  values  of 
)  X.    For  if  t/i  =  mxi,  and  y^  =  mx^  then  by  dividing  the  first 

equation  by  the  second,  we  obtain  th^  proportion  ^  =  — . 

I  2/2         X2 

j      The  phrase  "  variation  of  a  function  "  used  in  Section  12, 

I  should  not  be  confused  with  "  variation  "  as  a  heading  under 

i  which  we  discuss  proportional  variables.    A  more  general  use 

j  of  the  latter  term  will  be  taken  up  in  Chapter  III. 

I      The  mass  of  a  body  is  the  amount  of  matter  in  it.     If  the 

!  nature  of  the  material  in  the  body  is  the  same  throughout,  the 

I  mass  varies  as  the  volume.     That  is,  if  M  denotes  the  mass 

and  V  the  volume,  M  =  mV,  where  m  is  a  constant  called  the 

density  of  the  substance  of  which  the  body  is  composed.     The 

value  of  m  is  the  mass  of  a  unit  volume. 


62 


ELEMENTARY  FUNCTIONS 


Example.     If  the  mass  of  an  aluminum  object  is  13,  and  its  volume  is 
5,  find  the  mass  of  an  aluminum  object  whose  volume  is  8.6,  and  the  volume 
of  an  object  whose  mass  is  17.4. 

Let  M  denote  the  mass  and  V  the  volume  of  any 
aluminum  object.    Then  since  M  varies  as  V, 

M  =  mV.  (1) 

Graphical  solution.  The  graph  is  a  straight  line 
through  the  origin  (Corollary  1  to  Theorem  1,  Sec- 
tion 20).  It  also  passes  through  the  point  A  (5,  13), 
since  M  =  13  when  V  =  5.  It  is  therefore  the  line 
OA  in  the  figure. 

From  the  graph,  if  7=  8.6,  the  value  of  M  is 
CD  =  22.4;  and  if  M  =  17.4,  the  value  of  F  is  OG  = 
7.     These  are  the  required  values. 

The  slope  of  the  line,  computed  from  0  and  A,  is 


M' 

■ 

-22 

D/ 

^       ■■■ 

r/ 

E 

— 

— 

■'fl 

J  -L 

13 

-^ 

(i^ 

r 

-10 

'\   ! 

/ 

/ 

1 

^ 

?!   1 

0 

1 

ClfV 

Fig.  38. 


m  = 


AM      BA      13 


AV      OB 


^  =  ^   =  2.6. 


Since  -\^  is  the  mass  of  a  unit  volume  of  aluminum,  or  the  density, 
the  slope  represents  the  density. 

Algebraic  solution.  Substituting  the  given  values  M  =  13  and  V  =  5 
in  (1),  we  obtain  13  =  5m,  whence  the  density  m  =  ^^-  =  2.6. 

Substituting  the  value  of  m  in  (1),  we  have  the  relation  between  the  mass 
and  volume  of  any  aluminum  object, 

(2) 


Then  if  F  =  8.6, 
And  if  M  =  17.4, 
whence 


M  =  2.6F. 
Af  =  2.6x8.6  =  22.36. 
17.4  =  2.6F, 

V  =  17.4/2.6  =  6.69. 


The  graph  has  the  advantage  of  exhibiting  the  relation  to  the 
eye,  and  if  it  is  constructed  carefully,  it  is  sufficiently  accurate 
for  the  purpose  of  a  "  ready  reckoner/'  It  is  very  useful  when 
a  number  of  values  of  either  variable  are  desired  and  great  ac- 
curacy is  not  essential,  and  in  any  case  it  furnishes  a  valuable 
check  on  the  accuracy  of  the  computation. 

When  we  are  given  the  fact  that  a  variable  y  is  proportional 
to  a  second  variable  x,  the  general  form  of  the  law  connecting 
X  and  2/  is  2/  =  mx. '  The  exact  form  of  the  law  involves  a  par- 
ticular value  of  the  constant  m.  This  value  of  m  may  be  found 
from  a  pair  of  corresponding  values  of  x  and  y,  other  than  (0,  0), 
as  in  the  algebraic  solution  of  the  example  above 


LINEAR  FUNCTIONS  63 

22.  Uniform  Acceleration.  The  motion  of  a  body  is  said  to 
be  accelerated  if  the  velocity  is  variable.  It  is  said  to  be  uni- 
formly accelerated  if  the  change  in  velocity  is  proportional  to 
the  change  in  time.  The  rate  of  change  of  velocity  is  called 
acceleration,  and  it  is  measured  by  the  number  of  feet  per 
second  by  which  the  velocity  changes  in  each  second. 

A  freely  falhng  body  moves  with  a  uniformly  accelerated 
motion,  if  the  resistance  of  the  air  is  neglected.  The  accelera- 
tion, which  is  called  the  acceleration  due  to  gravity  and  which  is 
denoted  by  g,  is  approximately  32  feet  per  second  per  second. 
If  the  positive  direction  is  downward,  g  =  32,  but  if  the  positive 
direction  is  upward,  ^  =  —  32. 

Since  the  rate  of  change  of  velocity  of  a  uniformly  accelerated 
body  is  constant,  v  is  a  linear  function  of  t,  and  the  graph  is  a 
straight  line.  The  slope  of  the  hne  represents  the  acceleration, 
and  the  intercept  on  the  !;-axis  represents  the  velocity  when 
t  =  0,  which  is  called  the  initial  velocity. 

EXERCISES 

1.  Hooke's  law  says  that  the  amount  of  stretching  in  a  stretched  elastic 
string  is  proportional  to  the  tension.  If  a  2-lb.  weight  will  stretch  a  string 
3  feet,  what  tension  arises  when  the  string  is  stretched  1  foot?  Draw  the 
graph. 

2.  Prove  graphically,  that  if  y  varies  as  x,  then  any  two  values  of  y  are 
proportional  to  the  corresponding  values  of  x. 

3.  Is  the  definition  near  the  end  of  Section  16  equivalent  to  the  state- 
ment following?  A  variable  y  changes  uniformly  with  respect  to  x  if 
the  change  in  y  is  proportional  to  the  change  in  x  producing  it. 

4.  The  mass  of  a  body  varies  as  its  volume.  On  the  same  axes  con- 
struct the  graphs  showing  the  relation  between  mass  and  volume  for 
bodies  composed  of  the  following  substances  whose  densities  are  given: 

Lithium,  0.6;  alcohol,  0.8;  India  rubber,  1.0;  magnesium,  1.7;  diamond, 
3.5;  silver,  10.5.     How  are  the  various  densities  represented? 

5.  Wihelmy's  law  for  chemical  reactions  states  that  the  amount  of 
chemical  change  in  a  given  time  is  proportional  to  the  quantity  of  re- 
acting substance  in  the  system.     Construct  the  graph. 

6.  The  velocity  acquired  in  t  seconds  by  a  body  falling  freely  from  rest 
is  given  by  the  equation  v  =  d2t.  Plot  the  graph  and  from  it  determine 
how  fast  the  body  would  be  falling  after  4  seconds.  What  does  the  slope 
represent? 


64  ELEMENTARY  FUNCTIONS 

7.  If  a  ball  is  dropped  from  a  high  building,  how  fast  will  it  be  moving 
at  the  end  of  one  second?  at  the  end  of  2  seconds?  at  the  end  of  4  seconds? 
If  a  ball  is  thrown  vertically  upward,  how  will  its  velocity  be  affected  in 
any  second?  If  it  is  thrown  up  with  an  initial  velocity  of  96  feet  per 
second,  how  long  will  it  rise? 

8.  A  train  starting  from  rest  acquires  a  velocity  of  50  feet  per  second  in 
15  seconds.     Find  the  average  acceleration. 

9.  An  automobile  is  moving  at  the  rate  of  40  feet  per  second,  when  the 
power  is  shut  ofif  and  the  brakes  applied.  If  it  moves  thereafter  with  a 
uniform  acceleration  of  -  5  feet  per  second  per  second,  how  long  before 
it  will  stop? 

10.  Name  two  quantities  arising  in  every-day  experiences  which  are 
proportional,  and  give  the  value  of  the  constant  involved  in  the  relation 
between  them. 

11.  The  pressure  of  a  liquid  is  proportional  to  the  depth.  If  the  pres- 
sure per  square  inch  on  the  suit  of  a  diver  is  5.2  lbs.  for  a  depth  of  12  ft. 
how  deep  can  he  go  safely  if  78  lbs.  per  square  inch  is  an  allowable  pressure? 
Construct  the  graph. 

12.  For  small  changes  in  altitude,  atmospheric  pressure  varies  as  the 
altitude.  If  the  change  in  the  reading  of  a  barometer  is  0.1  of  a  unit  for 
each  90  ft.  ascent,  construct  the  graph  of  this  relation  and  determine 
the  difference  in  readings  of  two  places  with  a  difference  in  altitude  of 
1000  ft. 

13.  In  a  spring  balance  the  extension  of  the  spring  is  proportional  to 
the  weight.  If  a  weight  of  2  lbs.  lengthens  the  spring  1  inch,  construct  a 
graph  and  determine  the  extension  of  the  spring  for  weights  of  5,  8,  and 
17  pounds. 

14.  Thestockof  a  corporation  yields  an  annual  dividend  of  5%.  What 
is  the  relation  between  the  total  amount  a  stockholder  receives  in  divi- 
dends and  the  time  the  stock  is  held?  If  the  graph  of  this  relation  is 
drawn,  what  does  the  slope  of  the  line  represent?  What  is  the  value  of 
the  slope  if  the  par  value  of  the  stock  held  is  $10,000? 

15.  The  rent  charged  to  each  department  of  a  store  is  proportional  to 
the  percentage  of  the  total  floor  space  which  the  department  occupies. 
If  the  store  pays  a  rent  of  $12,000,  construct  a  graph  to  show  the  rent 
charged  to  a  department  of  any  size,  using  as  large  a  scale  as  convenient. 
From  the  graph,  determine  the  rent  charged  to  each  department  if  the 
various  departments  occupy  17,  25,  10,  8,  5,  12,  and  23  per  cent  of  the  total 
floor  space. 

16.  An  automobile  moving  at  the  rate  of  22  feet  per  second  (15  miles 
per  hour)  begins  to  coast  down  a  hill.  If  its  velocity  after  t  seconds  ia 
given  by  V  =  \Qt  +  22,  find  from  the  graph  of  v  how  fast  it  would  be  going 
after  5  seconds,  and  how  long  it  would  take  it  to  acquire  a  velocity  of  60 
miles  per  hour.    What  is  the  acceleration? 


LINEAR  FUNCTIONS 


65 


17.  If  a  ball  is  thrown  vertically  upward  with  a  velocity  of  100  feet 
per  second,  its  velocity  after  t  seconds  is  given  by  the  equation  v  =  -  32< 
+  100.  Plot  the  graph,  and  describe  the  motion.  What  is  represented  by 
the  intercept  on  the  v-axis?  the  intercept  on  the  <-axis?   the  slope? 

18.  The  ordinate  of  a  point  on  one  of  the  graphs  in  the  figure  repre- 
sents the  velocity  y  of  a  moving  body  at  the  time  t  represented  by  the 
abscissa.    Describe  the 

motion  of  a  body 
whose  velocity  is  repre- 
sented by  each  of  the 
graphs,  from  <  =  -  6 
to  <  =  12. 

Solution  for  (a). 
From  the  graph,  the 
intercept  on  the  i-axis 
is  f  =  6,  when  y  =  0,  so 
that  the  body  is  not 
moving  at  this  instant. 
The  intercept  on  the 
f-axis  is  y  =  -  2,  which 
gives  the  velocity  when 
<  =  0. 

The  line  is  below  the  <-axis  if  <  <  6,  so  that  during  this  time  v  is 
negative  and  the  body  is  moving  in  the  opposite  of  a  certain  direction 
which  has  been  assumed  as  positive,  i.e.,  in  the  negative  direction.  The 
line  is  above  the  <-axis  if  i  >  6,  so  that  v  is  positive  and  the  body  is  moving 
in  the  positive  direction. 

The  slope  of  the  line,  computed  from  the  points  (6,  0)  and  (0,  -  2)  is  the 

,      ^.  Ay      -  2  -  0      1  ., 

acceleration  m  =  ^  =  — — — -  =  -  ft.  per  sec.  per  sec. 


V. 

Vj 

V, 

(c) 

^ 

V, 

V 

V 

4 

b) 

— - 

_^ 

^ 

_,9 

K 

'^ 

^ 

— 

" 

.^ 

— 

■ — ' 

'v. 

"v. 

^ 

V 

^ 

^ 

-.' 

_.: 

I 

i><r 

-( 

-■ 

^ 

<: 

0 

: 

.^ 

i 

v^...p| 

:-i 

^ 

>- 

c 

^ 

V, 

L$ 

,^ 

^ 

^ 

"^ 

^ 

".^ 

^ 

(d) 

*^ 

-J 

Fig.  39. 


Since  the  line  rises  to  the  right,  the  velocity  constantly  increases,  which 
agrees  with  the  fact  that  the  acceleration  is  positive. 

From  the  graph,  when  t=  -  6,  y  =  -  4,  and  when  i  =  12,  y  =  2. 

Discarding  the  technical  terms  positive  and  negative,  the  motion  may 
be  described  as  follows :  At  the  start  the  body  is  moving  in  a  certain  direc- 
tion at  the  rate  of  4  feet  per  second.  During  the  next  12  seconds  it  slows 
down  uniformly  and  comes  to  rest  for  an  instant,  and  then  begins  to  move 
in  the  opposite  direction  with  a  constantly  increasing  speed.  After  6 
seconds  it  has  acquired  a  velocity  of  2  feet  per  second.  The  velocity 
changes  by  |  of  a  foot  per  second  in  each  second  of  the  motion. 

Notice  that  where  the  body  is  at  any  time,  and  in  particular  where  it 
comes  to  rest,  cannot  be  determined  from  the  graph  of  the  velocity. 

19.  The  velocity  of  an  automobile,  in  feet  per  second,  at  the  time  <, 


66  ELEMENTARY  FUNCTIONS 

in  seconds,  is  given  by  one  of  the  following  equations.     Construct  the 
graph,  and  describe  the  motion,  from  t  =  0  to  t  =  Q. 

(a)  V  =  U.         (b)  y  =  3<  -  6.        (c)  «  =  O.U  -  2.         (d)  w  =  -  2i  +  5. 

20.  Describe  the  motion  of  a  ball  rolling  on  the  side  of  a  hill  if  its  veloc- 
ity is  given  by  one  of  the  following  equations,  by  interpreting  the  graph. 
What  is  the  acceleration  in  each  case? 


(a)  V  =  10/. 

(b)  t;  =  8f  +  5. 

(c)  y  =  6«  -  8. 

(d)  y  =  -  4i  +  12. 

(e)  y  =  -  6i  +  12. 

(f)   t;  =  3^-12. 

21.  Construct  the  graph  of  the  equation  2/  =  -  2x  +  4.  From  it  de- 
scribe the  motion  of  a  body  (a)  if  y  represents  the  distance  of  a  body  from 
a  certain  station  at  the  time  x]  (b)  if  y  represents  the  velocity  of  a  body 
at  the  time  x. 

23.  Equation  of  a  Straight  Line.  We  have  seen  that  the 
graph  oi  y  =  mx  +  ?>  is  the  straight  line  whose  slope  is  m  and 
whose  intercept  on  the  2/-axis  is  h.  Conversely,  if  a  hne  is 
given  whose  slope  is,  say,  2  and  whose  intercept  on  the  2/-axis  is, 
say,  3,  then  the  equation  of  which  it  is  the  graph  is  ?/  =  2x  +  3; 
for  the  graph  of  this  equation  is  known  to  be  the  given  hne. 

The  equation  of  which  a  given  line  or  curve  is  the  graph  is 
called  the  equation  of  ihedine  or  curve. 

To  find  the  equation  of  a  hne  whose  slope  and  intercept  on 
the  2/-axis  are  given,  substitute  the  given  values  of  m  and  6  in 
the  equation 

y  =  mx  +  h.  (1) 

This  is  called  the  slope-intercept  form  of  the  equation  of  a 
straight  line. 

To  find  the  equation  of  a  line  determined  by  its  slope  and  any 
point  on  it  we  apply  the 

Theorem.  The  equation  of  the  line  whose  slope  is  m  and  which 
passes  through  the  point  Pi{xi,  yi)  is 

y  -yi  =  m(x  -  xi).  (2) 

Let  P(x,  y)  be  any  point  on  the  given  line.  Since  the  value 
of  Ay /Ax  for  any  two  points  on  a  straight  hne  is  constant 
(page  49),  and  equal  to  m  (definition  page  50),  we  have 


LINEAR  FUNCTIONS 
2/ -2/1 


67 


and  hence 


JSx      X  —  Xi 
2/  -  2/1  =  m{x 


=  w, 


xr). 


This  is  called  the  point-slope  form  of  the  equation  of  a 
straight  Une. 

To  ^nd  the  equation  of  a  straight  line  determined  by  two  points^ 
find  the  slope  and  then  apply  (2). 

Example  1.    Find  the  equation  of  the  line  determined  by  the  points 
A  (1,  4)  and  B  (2,  3). 
The  slope  oi  AB'm 

3-4         , 

Substituting  xi  =  1,  t/i  =  4,  and  m  =  -  1  in  (2) 
we  get 

2/  -  4  =  -  l(x  -  1), 
or  X  + 1/  -  5  =  0.  Fig.  40. 

Check.  The  coordinates  of  A  and  of  B  satisfy  this  equation. 

Example  2.  Find  the  equation  of  the  hne 
through  the  point  A{b,  2)  which  is  parallel  to 
the  Hne  x  -  3?/  +  6  =  0. 

Solving  the  given  equation  for  y  we  get 


K 

1 

" 

\ 

A  (1,4^ 

^ 

\ 

\ 

B(3,S 

-1 

\ 

- 

\ 

0 

. 

1     1 

'!  t1 

i/ 

,<?^^ 

ri-^ 

"^ 

? 

^ 

^( 

5,2)^ 

,^ 

^ 

^ 

"^""^ 

u; 

:::5''^ 

0 

. 

5:^ 

J 

2/  =  Ix  +  2, 

and  hence  the  slope  of  the  given  line  is  m  =  |, 
the  coefficient  of  x  (Corollary  2,  to  Theorem  2, 
Section  20).    Hence  the  slope  of  the  required  y\q.  41. 

Une  is  \  (Theorem  2,  Section  18). 
Then  by  (2)  the  required  equation  is 

y-1  =  \{x-  5). 
or  a;  -  3t/  +  1  =  0. 

Example  3.  The  freezing  point  on  a  Centigrade  thermometer  is  0°, 
and  on  a  Fahrenheit  thermometer  it  is  32°.  The  boiling  point  on  the 
first  is  100°  and  on  the  second  is  212°.  Find  the  relation  between  any 
two  corresponding  readings  on  the  two  thermometers. 

Since  the  interval  between  the  freezing  and  boiling  points  Is  divided 
into  equal  parts  on  each  thermometer,  a  change  of  one  degree  on  one  of 
them  will  always  produce  a  definite,  constant  change  on  the  other.  Hence 
the  graph  will  be  a  straight  line. 


68 


ELEMENTARY  FUNCTIONS 


In  plotting,  let  abscissas  denote  readings  on  the  Centigrade  scale,  and 
ordinates  the  corresponding  readings  on  the  Fahrenheit  scale.     Then  the 

readings  for  the  freezing  and 
^^  boihng  points  are  represented 

respectively  by  the  points 
Pi(0,  32)  and  PzClOO,  212). 
The  required  relation  is  the 
equation  of  the  straight  Une 
P1P2. 
The  slope  of  the  line  is 


PJ100.212) 


10  20  so  40  50  60  70  80  90 100    Q 

Fig.  42. 


m 


212  -  32 


100  "  ■^•^• 


100-  0 
The  intercept  on  the  P-axis 


is  6  =  32.     Hence,  using  (1),  the  relation  is 

F  =  1.8C  +  32. 

The  slope  of  the  line,  1.8,  is  the  rate  of  change  of  a  reading  on  the 
Fahrenheit  scale  with  respect  to  the  Centigrade  scale.  As  the  rate  of 
change  is  measured  by  the  change  on  the  Fahrenheit  scale  due  to  a  unit 
change  on  the  Centigrade  scale,  this  means  that  1°.8  Fahrenheit  =  1° 
Centigrade. 


EXERCISES 

1.  Construct  each  of  the  lines  indicated  below.  Find  its  equation  and 
reduce  it  to  the  form  Ax  +  By  -^  C  =  0.  Check  the  result  by  substitut- 
ing the  coordinates  of  the  given  point  or  points. 

(a)  With  the  slope  \  and  the  intercept  on  the  2/-axis  -  4. 

(b)  Passing  through  the  point  (3,  2)  with  the  slope  -  |. 

(c)  Determined  by  the  points  (2,  5)  and  (-  1,  2). 

(d)  Through  the  point  (3,  -  4)  parallel  to  the  x-axis;  the  y-axis. 

(e)  Passing  through  the  points  (0,  4)  and  (3,  0). 

(f)  Through  the  point  (1,  3)  parallel  to  the  line  2/  =  2x  -  4. 

(g)  Through  the  point  (3,  0)  parallel  to  the  line  determined  by  the 
points  (1,  2)  and  (5,  4). 

(h)  Through  the  point  (-2,  -  3)  parallel  to  the  line  x  +  2t/  -  6  =  0. 

2.  What  is  the  general  form  of  the  equation  of  a  line  parallel  to  the 
X-axis?  the  2/-axis? 

3.  What  is  the  value  of  A  if  the  line  Ax  -  3!/  +  5  =  0  is  parallel  to  the 
line  3x  +  4i/  -  12  =  0?    If  its  intercept  on  the  x-axis  is  10? 

4.  A  medicine  is  40  %  alcohol.  Construct  a  graph  showing  the  amount 
of  alcohol  in  any  amount  of  the  medicine,  and  find  its  equation. 

6.   In  an  experiment  in  stretching  a  brass  wire,  it  is  assumed  that  th( 
elongation  E  is  connected  with  the  tension  T  by  a  linear  relation.    If' 


LINEAR  FUNCTIONS 


69 


i 


T  =  18  pounds  when  E  =  0.1  inch,  and  T  =  58  pounds  when  E  =  0.3 
inches,  construct  the  graph  and  find  the  relation. 

6.  The  population  of  a  town  in  1900  was  1200  and  in  1910  it  was  1400. 
Assuming  that  the  growth  in  population  was  uniform,  construct  the 
graph  showing  the  population  at  any  time,  and  find  the  relation  between 
the  population  and  the  time. 

7.  A  sum  of  money  at  simple  interest  amounts  to  $94.00  in  5  years,  and 
$108.00  in  10  years.     Find  the  sum  of  money  and  the  rate  of  interest. 

8.  It  is  observed  that  the  boiling  point  of  water  at  sea  level  is  212**  F., 
and  that  at  an  altitude  of  2299  feet  it  is  208**  F.  What  is  the  boiUng 
point  at  an  altitude  of  5000  feet? 

9.  Find  the  coordinates  of  the  point  of  intersection  of  each  of  the  pairs 
of  lines  given  below. 

(a)  a;  +  2/  =  5,  and  x  -  y  =  I. 

Let  A  be  the  point  of  intersection.  Then 
since  A  is  on  both  graphs,  its  coordinates 
must  satisfy  both  equations,  and  hence  these 
coordinates  may  be  found  by  solving  the 
equations  simultaneously. 

Adding  the  equations,  we  get  2x  =  6, 
whence  x  =  3. 

Substituting  in  either  equation  we  find  that 
y  =  2.  Therefore  the  coordinates  of  A  are 
(3,  2). 

(b)  X  +  2i/  -  6  =  0  and  a;  -  2/  +  3  =  0. 

(c)  3a;  +  7/  +  3  =  0  and  X  -  3?/  -  5  =  0. 

(d)  0.3x  -  0.8?/  +  5.3  =  0,  and  y  =  1.5a:. 

10.  Find  the  coordinates  of  the  point  of  intersection  of  the  line  de- 
termined by  the  points  (2,  1)  and  (4,  3)  and  the  line  through  the  point 
(1,  5)  whose  slope  is  -  |. 


v 

<            ^ 

.%l     ^ 

s^.^^ 

^z 

zs 

-,^     s     ^ 

■J  0 /.   ,             \i   ri 

} 

Fig.  43. 


24.  Application  to  the  Solution  of  Problems.  In  the  fol- 
lowing examples  and  exercises  it  is  seen  how  the  graphical 
solution  of  problems  solvable  by  functions  which  change  imi- 
formly  may  be  used  to  suggest  a  method  of  algebraic  solution. 


Example  1.    Solve  Example  1,  Section  19,  algebraically. 

The  graphical  solution,  page  54,  shows  that  the  value  of  DU  must  be 
found.  But  DE  =  OH  -  OD.  It  is  known  that  OD  =  3,  and  since  OE 
is  the  abscissa  of  G,  it  may  be  found  by  solving  the  equations  of  OC  and 
EF  for  X. 

Since  OC  passes  through  the  origin  and  has  the  slope  0.2,  its  equation  is 

y  =  0.2a;.  (1) 


70  ELEMENTARY  FUNCTIONS 

The  abscissa  of  ^  is  x  =  3,  and  since  E  lies  on  the  line  OB,  whose  equa- 
tion is  2/  =  0.1a;,  its  ordinate  is  ?/  =  0.1  x  3  =  0.3.  Hence  the  coordinates  of 
E  are  (3,  0.3).  Since  EF  was  drawn  parallel  to  OA  its  slope  is  m  =  0.5, 
and  therefore  its  equation  has  the  form  (Theorem,  Section  23) 

2/  -  0.3  =  0.5(x  -  3), 
or  y  =  0.5x  -  1.2.  (2) 

To  find  OH,  the  abscissa  of  G,  solve  (1)  and  (2)  for  x.  Eliminating  y, 
we  have 

0.2x  =  0.5x  -  1.2, 
whence  O.Sx  =  1.2, 

and  therefore  OH  =  x  =  4. 

Then  DH  =  OH  -OD  =  4-S  =  1. 

Example  2.     Solve  Example  2,  Section  19,  algebraically. 

The  time  of  the  freight  train  and  the  distance  from  A  to  B  are  repre- 
sented by  the  coordinates  of  the  point  G  (Figure  35,  page  55),  the  point 
of  intersection  of  the  lines  OC  and  FG.  The  equations  of  these  lines  are 
found  to  be  respectively 

OC  :  d  =  15t, 
and  FG:  d  =  45<  -  202.5. 

Solving  these  equations  we  get 

/  =  6|  hours,  the  time  of  the  freight;  and 
d  =  101|  miles,  the  distance  AB. 

The  time  of  the  express  train  is  represented  by 

DI  =  FJ  =  0J  -OF  =  &l-^  =  2\  hours. 

EXERCISES 

Solve  each  of  the  exercises  below  both  graphically  and  algebraically 
using  the  graphic  solution  as  the  basis  of  the  algebraic  solution. 

1.  Two  trains  start  toward  each  other  from  Buffalo  and  New  York, 
respectively,  440  miles  apart.  The  one  from  New  York  travels  at  the 
rate  of  50  miles  an  hour,  and  the  one  from  Buffalo  at  the  rate  of  40  miles 
an  hour.    When  and  where  will  they  meet? 

2.  A  man  walks  at  the  rate  of  3  miles  an  hour.  Three  hours  after  he 
starts,  another  man  starts  from  the  same  place  and  travels  in  the  same 
direction  at  the  rate  of  10  miles  per  hour.  When  and  where  will  the  latter 
overtake  the  former? 

3.  An  express  train  starts  from  a  city  and  moves  with  a  velocity  of  40 
miles  per  hour.  A  freight  train  is  90  miles  ahead  of  the  express  at  the 
start  and  is  moving  at  the  rate  of  25  miles  an  hour.  When  and  where 
will  the  express  overtake  the  freight? 


LINEAR  FUNCTIONS  71 

4.  When  and  where  will  the  express  train  in  Exercise  3  be  30  miles  be- 
hind the  freight? 

5.  Two  trains  leave  a  city  at  the  same  time,  traveling  at  the  rates  of 
30  and  45  miles  an  hour  respectively.  One  arrives  at  another  city  3  hours 
ahead  of  the  first.  Find  the  distance  between  the  cities  and  the  time  in 
which  each  train  made  the  trip. 

6.  How  much  vinegar  must  be  added  to  a  barrel  of  vinegar  60  %  pure 
to  make  it  an  80%  solution? 

7.  A  cough  medicine  is  50  %  paregoric.  How  much  paregoric  should 
be  added  to  4  ounces  of  the  medicine  to  make  it  75  %  paregoric? 

8.  Brass  is  an  alloy  of  copper  and  zinc.  How  many  cubic  centimeters  of 
zinc,  density  6.9,  must  be  combined  with  100  cubic  centimeters  of  copper, 
density  8.8,  to  form  brass  whose  density  is  8.3? 

9.  Coinage  silver  is  an  alloy  of  copper  and  silver.  How  many  cubic 
centimeters  of  copper,  density  8.8,  must  be  added  to  10  cubic  centimeters 
of  silver,  density  10.6,  to  form  coinage  silver  whose  density  is  10.4? 

10.  At  what  time  between  2  and  3  are  the  hands  of  a  clock  together? 
opposite? 

11.  At  what  time  between  7  and  8  are  the  hands  of  a  clock  together? 
opposite? 

12.  In  a  clock  which  is  not  keeping  true  time,  it  is  observed  that  the 
interval  between  successive  coincidences  of  the  hour  and  minute  hands  is 
66  minutes.     What  is  the  error  of  the  clock? 

13.  The  planet  Merciury  makes  a  circuit  around  the  sun  in  3  months, 
and  Venus  in  7^  months.  Find  the  time  between  two  successive  times 
when  Mercury  is  between  Venus  and  the  sun. 

14.  A  shoe  dealer  buys  100  pairs  of  shoes  at  $2.00  a  pair,  and  sells  75 
pairs  at  $2.50  a  pair.  To  sell  the  remaining  shoes  he  marks  them  down 
80  as  to  make  20  %  profit  on  the  whole.  What  price  per  pair  will  give  this 
result? 

16.  A  and  B  can  do  a  piece  of  work  in  10  days,  but  at  the  end  of  7  days 
A  falls  sick  and  B  finishes  the  work  in  5  days.  How  long  would  it  take 
each  man  to  do  the  work? 

16.  One  man  can  do  a  piece  of  work  in  5  days  which  it  takes  a  second 
man  7  days  to  do.     How  long  will  it  take  the  two  men  working  together? 

17.  A  piece  of  work  can  be  done  by  A  in  10  days,  and  by  B  in  12  days. 
If  A  starts  the  work  and  works  alone  for  4  days,  how  long  will  it  take  A 
and  B  working  together  to  complete  it? 

18.  A  pound  of  a  certain  alloy  contains  two  parts  of  silver  to  three 
parts  of  copper.  How  much  copper  must  be  melted  with  this  alloy  to 
obtain  one  which  contains  three  parts  of  silver  to  seven  of  copper? 

19.  When  weighed  in  water,  silver  loses  0.09  of  its  weight,  and  copper 
0.11  of  its  weight.  If  a  12-pound  mass  of  silver  and  copper  loses  1.17 
pounds,  find  the  weight  of  the  silver  and  of  the  copper  in  the  mass. 


72  ELEMENTARY  FUNCTIONS 

20.  The  crown  of  Hiero  of  Syracuse,  which  was  part  gold  and  part 
silver,  weighed  20  pounds,  and  lost  1^  pounds  when  weighed  in  water. 
How  much  gold  and  how  much  silver  did  it  contain,  if  19?  pounds  of  gold 
and  10^  pounds  of  silver  each  lose  1  pound  in  water? 

21.  The  admission  to  an  entertainment  was  50ff  for  adults  and  25^  for 
children.  The  proceeds  from  125  tickets  were  $51.25.  How  many  adults 
and  how  many  children  were  admitted? 

22.  A  tank  can  be  filled  by  one  pipe  in  20  minutes,  and  by  another 
pipe  in  30  minutes.  How  long  will  it  take  to  fill  the  tank  if  both  pipes  are 
opened? 

23.  A  tank  can  be  filled  by  one  pipe  in  5  hours  and  emptied  by  another 
in  8  hours.  If  the  tank  is  half  full,  and  both  pipes  are  opened,  how  long 
will  it  take  to  fill  the  tank? 

25.  Remarks  on  Measurements.  This  section  and  the  one 
following  contain  considerations  which  are  of  value  to  all  who 
make  measurements  and  computations  based  upon  them. 
In  particular,  they  are  important  for  the  statistician.  Some 
of  them  find  application  in  connection  with  tables  of  values  of 
several  functions  which  we  shall  study. 

The  operation  of  making  a  measurement  is  counting.  It 
is  the  determination  of  the  number  of  units  of  a  certain  kind 
required  to  equal  a  magnitude  of  the  same  kind. 

In  measuring  a  length  with  a  scale  graduated  to  tenths  of 
an  inch,  the  length  is  recorded  to  the  nearest  tenth.  For 
instance,  if  a  line  appears  to  be  7.8|  inches,  the  length  is  re- 
corded as  7.8  inches;  if  it  is  about  7.8f  inches  it  is  recorded  as 
7.9  inches.  If  the  length  is  so  near  the  mid  point  of  a  sub- 
division that  it  is  impossible  for  the  observer  to  decide  which 
point  of  division  is  nearer  it  is  customary  to  choose  the  sub- 
multiple  which  is  even.  Thus  7.8J  inches  is  recorded  as  7.8 
inches,  and  7.3J  as  7.4  inches.  An  experienced  observer  de- 
termines the  length  to  the  hundredth  part  of  an  inch  by  making 
a  mental  subdivision  of  the  tenth  of  an  inch.  If  a  higher 
degree  of  accuracy  is  desired,  instruments  which  measure  more 
precisely  are  employed,  and  various  indirect  methods  of 
measurement  are  used. 

If  a  change  of  units  is  made  the  figures  of  a  measurement 
may  be  preceded  or  followed  by  ciphers  which  are  not  deter- 


LINEAR  FUNCTIONS  73 

mined  by  observation.  Thus  the  same  length  may  be  re- 
corded as  0.054  meter,  54  milUmeters,  or  54,000  microns. 
The  digits  5,  4  are  called  the  significant  figures.  The  ciphers 
are  non-significant. 

Ciphers  in  a  number  are  not  always  non-significant.  A 
length  between  the  limits  7.795  inches  and  7.805  would  be 
recorded  as  7.80  inches. 

The  distance  to  the  sun  is  sometimes  given  as  93,000,000 
miles.  This  statement  is  ambiguous  as  there  is  nothing  to 
indicate  how  many  of  the  figures  are  significant  and  how 
many  of  the  ciphers  merely  serve  to  locate  the  decimal  point 
with  reference  to  the  unit  employed.  If  all  the  ciphers  were 
significant  then  the  statement  would  mean  that  the  distance 
was  between  the  limits  92,999,999.5  and  93,000,000.5  miles. 
If  the  figures  9,  3  alone  were  significant,  then  the  limits 
would  be  92,500,000  and  93,500,000  miles.  In  a  notation  used 
to  remove  such  ambiguities,  known  as  the  standard  form,  the 
sun's  distance  given  to  two  significant  figures  would  be  written 
9.3  X  10^  miles.  If  the  first  three  figures  were  significant  the 
distance  in  standard  form  would  be  9.30  x  10^,  which  would 
indicate  that  the  true  distance  is  between  92,950,000  and 
93,050,000  miles. 

A  number  like  ir  or  \/2  can  be  calculated  to  any  degree  of 
accuracy  desired,  but  there  is  a  limit  to  the  number  of  signifi- 
cant figures  which  can  be  obtained  by  measurement. 

The  relative  error  in  a  measurement  is  the  ratio  of  the  possible 
error  to  the  measurement.  It  is  usually  expressed  as  a  per- 
centage. 

^  The  relative  errors  in  the  two  distances  9.3  X  10^  and 
9.30  X  10^  are  respectively 

^  =  0.005  =  0.5  %      and     ^^  =  0.0005  =  0.05  %. 

This  illustrates  the  fact  that  the  relative  error  depends  upon 
the  number  of  significant  figures  and  not  upon  the  position  of 
the  decimal  point.  Let  x  be  a  number  obtained  by  measure- 
ment, expressed  in  terms  of  the  smallest  unit  used.    Then  x 


74  ELEMENTARY  FUNCTIONS 

differs  from  the  true  value  by  not  more  than  |,  and  the  possible 

1        -j 
relative  error  is  -  =  ^  •     The  possible  relative  error  for  dif- 

ferent  numbers  of  significant  figures  is  given  in  the  following 
table. 

Number  of  sig-  Range  of 

nificant  figures  numbers  Range  of  possible  relative  error. 

One  1,  .  .  . ,  9  2  *^  18  ^^  ^^  ^*'  **^  ^-^^  ^*' 

Two  10,  ...,99  i;  torpor  5%  to  0.505% 


Three  100,  ... ,  999  2^  *«  i^  <^^  ^.5  %  to  0.050  % 

Four  1000,  .  .  .  ,  9999         ^  to  :r^  or  0.05  %  to  0.005  % 


20      198 

J_.       1 

200^1998 

1  1 

2000  °  19998 


The  illustrations  following  show  how  great  an  accuracy  has 
been  obtained  in  some  measurements.  The  ratio  of  the  mean 
solar  day  to  the  sidereal  day  has  been  ascertained  to  the  eighth 
place  of  decimals.  Balances  have  been  constructed  which 
respond  to  one  part  in  a  million.  The  accuracy  attained  in 
measuring  a  base  line  of  a  survey  is  usually  about  one  part  in 
60,000,  or  an  inch  in  a  mile,  but  accuracy  to  one  part  in  a  mil- 
lion is  claimed  for  some  surveys. 

Three  significant  figures  are  sufficient  for  most  engineering 
and  commercial  calculations,  four  significant  figures  for  most 
physical  and  chemical  computations,  while  some  astronomical 
and  geodetic  calculations  require  measurements  to  six  or  seven 
significant  figures. 

26.  Possible  Errors  in  Arithmetic  Calculations.  Abridged 
Multiplication  and  Division.  As  magnitudes  determined  by 
measurements  are  not  exact,  it  is  important  to  be  able  to  esti- 
mate the  possible  error,  or  the  limit  of  error,  in  a  result  cal- 
culated from  such  measurements. 

Theorem  1.  The  possible  error  of  the  sum  or  difference  of 
two  measurements  is  equal  to  the  sum  of  the  possible  errors  of  the 
individual  measurements. 

If  a  and  b  are  two  measurenjents  with  possible  errors  =«=Aa 


LINEAR  FUNCTIONS  75 

and  =fcA6,  then  the  correct  value  of  the  sum  of  the  measure- 
ments lies  between  the  Hmits 

(a  +  Aa)  +  (6  +  A6)  =  (a  +  6)  +  (Aa  +  A6), 
and  (a  -  Aa)  +  (6  -  Ab)  =  (a  +  6)  -  (Aa  +  Ab). 

Hence  the  possible  error  of  the  sum  a  +  6  is  Aa  +  Ab. 

The  case  for  the  difference  of  two  measurements  follows  from 
the  fact  that  the  correct  value  of  the  difference  lies  between  the 
limits 

(a  +  Aa)  -  (6  -  Ab)  =  (a  -  6)  +  (Aa  +  A6), 
and  (a  -  Aa)  -  (6  +  Ab)  =  (a  -  b)  -  (Aa  +  Ab). 

Notice  that  in  either  case  the  possible  error  is  half  the  dif- 
ference of  the  Unaits. 

If  several  numbers  representing  measurements  of  different 
degrees  of  precision  are  added  together,  the  significant  figures 
are  retained  in  the  numbers  and  the  sum  is  rounded  off  to  the 
number  of  significant  figures  in  the  least  accurate  of  the  in- 
dividual numbers. 

Example  1.  The  value  of  the  imports  of  Alaska  from  the  United 
States  in  1901,  1903,  1904  are  reported  as  in  the  table.     Find  the  sum. 

The  first  number  is  rounded  off  to  thousands  so  that  the  sum  should 
be  rounded  ofiE  to  thousands.    This  may  be  done  as  the  numbers  are  added. 

$13,457,000 
9,509,701 
10,165,110 
$33,132,000 

Theorem  2.     The  possible  relative  error  of  the  product  or 

■  quotient  of  two  measurements  is  approximately  equal  to  the  sum 
of  the  relative  errors  of  the  individuM  measurements. 

The  product  will  lie  between  the  limits 

(a  +  Aa)  (6  +  Ab)  =  ab  +  AaAb  +  aA6  +  6 Aa, 
\  and  (a  -  Aa)  (6  -  Ab)  =  ab  +  AaAb  -  aAb  -  6 Aa. 

The  possible  error  in  the  prodi^ct  is  assumed  to  be,  approxi- 
mately, half  the  difference  between  these  limits,  so  that  the 

■  error  in  the  product  is 

Aa6  =  aA6  +  6Aa. 


76  ELEMENTARY  FUNCTIONS 

Hence  the  possible  relative  error,  that  is,  the  ratio  of  the 
possible  error  to  the  product,  is 

Aab  _  A6      Aa 
ab        b        a 

The  relative  error  of  ab  is  therefore  equal  to  the  sum  of  the 
relative  errors  of  a  and  b. 

The  case  for  the  quotient  is  proved  similarly,  assuming  that 
the  possible  error  is  half  the  difference  of  the  limits 

a  +  Aa  ,        a  —  Aa 

rrA6        ^"""^        6TA6' 

with  the  further  assumption  that  (AbY  is  small  in  comparison 
with  Aa  and  Ab  and  may  be  neglected. 

Since  the  possible  relative  error  of  a  product  is  greater  than 
that  of  either  factor,  there  can  be  no  more  significant  figures 
in  the  product  than  in  the  factor  with  the  smaller  number  of 
significant  figures.*  In  practice,  factors  are  rounded  off  to 
the  same  number  of  significant  figures  before  multiplying,  and 
only  that  number  of  significant  figures  retained  in  the  product. 

In  division,  the  dividend,  divisor,  and  quotient  are  treated 
similarly. 

Abridged  methods  of  multiplication  and  division,  which  do 
away  with  the  labor  of  retaining  non-significant  figures  during 
either  operation,  are  illustrated  in  the  examples  following. 

Example  2.  Find  the  circumference  of  a  circle  whose  measured  diam- 
eter is  45.88  inches. 

Since  the  diameter  is  given  to  four  significant  figures,  the  value  of  tt 
should  be  taken  to  four  significant  figures.    This  value  is  tt  =  3.142. 

(a)         45.38  (b)    45.38  (c)    40.00 

3.142  3.142  3.142 


I 


9 

181 

453 

13614 


142.58 


076  13614 

52  453 

8  181 

9 


13614 
8  454 

52  1 

076  

396  142.59 


396  142.58 

The  value  of  the  circumference,  wd,  to  four  significant  figures  is  therefore 
142.6. 

*  There  may  be  less.     For  example,  if  two  factors  with  three  signifi- 
cant figures  are  close  to  100,  such  as  123  and  114,  the  possible  relative 


LINEAR  FUNCTIONS  77 

Using  ordinary  multiplication  first  we  have  (a),  which 
gives  the  arithmetically  correct  product.  The  last  partial 
product  contributes  most  to  the  significant  figures  of  the 
product,  and  may  be  written  first,  as  in  (b),  where  the  order  of 
all  the  partial  products  is  reversed.  The  figures  to  the  right 
of  the  vertical  lines  in  (a)  and  (b)  pertain  to  the  non-significant 
figures  of  the  product,  and  may  be  discarded,  as  in  (c),  which 
shows  the  abridged  multiplication. 

In  (c),  the  partial  product  obtained  from  the  first  figure  on 
the  left  of  the  multipHer,  3,  is  entered  first.  Then  the  last 
number  on  the  right  of  the  multiplicand,  8,  is  canceled,  and 
the  second  figure  of  the  multiplier,  1,  is  used,  the  amount 
carried  over  from  the  canceled  figure  being  added  in.  The 
other  partial  products  are  rounded  off  similarly. 

Example  3.  Find  the  density  of  a  mass  of  7.643  grams  whose  volume 
Is  3.564  cubic  centimeters. 

To  obtain  the  density  divide  the  mass  by  the  volume.  As  there  are 
four  significant  figures  in  both  dividend  and  divisor,  neither  need  be 
rounded  off  before  the  division  is  performed,  and  the  quotient  should  be 
obtained  to  four  significant  figures. 

2.144  2.144 


3.564 1  7.643000  ^  3.  m4\  7.643 

7128  7128 

5150  515 

3564  356 

15860                                    •  159 

14256  143 

16040          •  16 

14256  14 

Ordinary  long  division,  given  on  the  left,  shows  that  the 
r  density  is  2.144. 

I  The  abridged  division  is  shown  on  the  right.  After  the 
ii  first  figure  in  the  quotient  is  obtained,  instead  of  adding  a 
;  cipher  to  the  dividend,  the  last  figure  of  the  divisor,  4,  is  can- 
'^  celed.     The  next  partial  product  is  rounded  off  in  accordance 

(   error  in  each  is  about  0.5  %  (by  the  table  in  the  preceding  section),  and  the 
li  possible  relative  error  in  the  product  is  therefore  approximately  1  %, 
I  which  indicates  that  only  two  significant  figures  should  be  retained  in  the 
product. 


78  ELEMENTARY  FUNCTIONS 

with  the  amount  carried  over  from  the  multipHcation  of  the 
canceled  number  by  the  second  digit  in  the  quotient,  etc. 
Ordinarily,  in  finding  the  partial  products  it  is  sufficient  to 
consider  only  the  nearest  canceled  figure  of  the  divisor,  but  it 
is  sometimes  necessary  to  consider  two  canceled  figures  to 
determine  the  amount  to  be  carried  over. 


EXERCISES 

1.  Write  the  following  numbers  in  standard  form  and  determine  the 
number  of  significant  figures  and  the  percentage  of  error  in  each:  3.1416, 
0.00732,  259.34,  678943,  0.0020. 

2.  The  value  of  tt  to  nine  figures  is  3.14159265.  Round  off  the  value 
to  one  significant  figure;  two;  three;  four;  five.  Determine  tlje  number  of 
significant  figures  and  the  percentage  of  error  of  the  approximations  3^ 
and  fit .  Correct  such  of  the  following  rounded  off  values  of  tt  as  are 
incorrect:  3.141,  3.15,  3.14160. 

3.  Add  the  following  numbers  obtained  by  measurements:  3.4785, 
16.743,  253.78,  36.583. 

4.  The  dimensions  of  a  rectangle  found  by  measurement  are  24.78 
inches  and  19.8  inches.  Find  the  area  and  determine  the  number  of 
significant  figures  and  the  relative  error  in  the  area. 

5.  The  diameter  of  a  circle  is  found  by  measurement  to  be  151.6  mil- 
limeters. Find  the  circumference.  Is  3f  a  sufficiently  accurate  approxima- 
tion for  TT  in  this  instance? 

6.  If  one  square  meter  is  equivalent  to  1.196  square  yards,  to  how  much 
are  5  square  meters  equivalent?  If  5  square  meters  are  equivalent  to  5.98 
square  yards,  to  what  is  one  square  meter  equivalent? 

7.  If  1  centimeter  is  equivalent  to  0.39^7  inches,  convert  a  measure- 
ment of  3.85  inches  to  centimeters.     Convert  42.83  centimeters  to  inches. 

8.  If  the  length  of  the  year  is  365.24  days  and  the  average  distance  of 
the  earth  from  the  sun  is  9.31  x  10^  miles,  find  approximately  the  velocity 
of  the  earth  in  miles  per  hour,  assuming  that  the  orbit  is  a  circle  with  the 
sun  at  the  center. 

27.  Empirical  Data  Problems.  It  is  frequently  possible  to 
measure  corresponding  pairs  of  values  of  two  related  variables 
when  the  law  connecting  them  is  unknown.  An  important 
problem  presented  by  such  empirical  data  is  to  determine  a  law 
which  represents  approximately  the  relation  between  the  two 
variables. 


LINEAR  FUNCTIONS 


79 


If  the  pairs  of  values  be  plotted,  it  may  happen  that  the 
points  so  obtained  he  very  nearly  on  a  straight  line.  If  such  is 
the  case,  it  is  reasonable  to  assume  that  the  graph  of  the  relation 
is  a  straight  line,  and  hence  that  the  relation  connecting  the 
two  variables  is  a  linear  equation. 

When  graphical  methods  are  being  used,  all  that  is  needed  is 
the  graph  of  the  relation.  Any  straight  Une  which  passes 
through  or  near  to  each  of  the  points  will  serve  approximately 
as  the  graph.  A  method  frequently  used  by  engineers  to  get 
the  line  giving  the  best  approximation  is  to  stretch  a  rubber 
band  over  two  pins  stuck  in  the  drawing  board,  and  move  the 
pins  about  until  the  stretched  band  appears  to  make  the  average 
distance  from  the  band  to  each  of  the  points  as  small  as  possible. 

When  algebraic  methods  are  employed,  a  method  of  obtain- 
ing the  equation  of  a  hne  which  answers 
well  as  the  graph  is  illustrated  in  the  p 
following  examples.  *- 

Example  1.  In  an  experiment  dealing  with 
friction  of  wood  on  wood,  a  block  of  wood  with 
various  weights  on  it  was  placed  on  a  hori- 
zontal board.  A  string  fastened  to  the  block 
ran  over  a  pulley  at  the  end  of  the  board,  and 

a  pan  was  tied  to  the  hanging  end.  Weights  were  placed  in  the  pan 
until  the  block  was  just  on  the  point  of  moving.  Using  W  to  represent 
the  combined  weight  of  the  block  and  the  weights  on  it,  and  W  for  the 

weight  of  the  pan  and  of  the 
faoM.2)    weights     in     it,     corresponding 
values  of  W  and  W  were  found 
as   indicated   in   the  table, 


W  >- 


Fig.  44. 


W 


10,        20, 


30. 


40 


W 


3.1,       5.8,       8.1,      11.2' 


the  unit  of  weight  being  the  gram. 
Determine  approximately  the  re- 
lation between  W  and  W. 
Fig.  45.  Plot  the  points  whose  coordi- 

nates are  the  pairs  of  values  of 
W  and  W,  using  values  of  W  as  abscissas.  The  four  points  obtained  lie 
very  nearly  on  a  straight  line  through  the  origin,  and  hence  we  assume 
that  the  graph  of  the  relation  is  a  straight  line  through  the  origin. 


80 


ELEMENTARY  FUNCTIONS 


(That  the  graph  ought  to  pass  through  the  origin  is  clear,  for  if  TT  =  0, 
so  also  must  W  =  0.)  The  required  relation  must  therefore  be  of  the 
form  W  =  mW,  where  m  is  the  slope  of  the  line. 

The  slopes  of  the  lines  joining  the  origin  to  each  of  the  four  points  are 
respectively 


3.1  -0 
10-0 


0.31; 


5^ 
20 


0.29; 


8J. 
30 


=  0-27;    ^-^  =  0.28. 


From  these  values  it  appears  that  m  may  have  any  value  between  0.27 
and  0.31,  and  the  line  W  =  mW  would  be  a  fair  approximation  to  the  graph 
required.  We  shall  choose  as  a  good  value  of  m  the  average  of  the  four 
values  of  m,  namely, 


m  =  ^  (0.31  +  0.29  +  0.27  +  0.28)  =  —■ 


0.29. 


In  dividing  1.15  by  4,  the  second  decimal  figure  is  8,  but  the  quotient 
Is  nearer  to  0.29  than  to  0.28;  hence  the  former  value  is  chosen. 

Hence  the  relation  desired  is  represented  approximately  by  the  equation 

W  =  0.29Tr.  (1) 

The  accuracy  with  which  this  equation  represents  the  given  data  may 

be  determined  by  constructing 
the  table  adjoined.  The  first 
two  columns  give  the  observed 
values  of  W  and  W,  the  third, 
the  values  of  W  computed  by 
means  of  equation  (1)  from  the 
observed  values  of  W]  the 
fourth,  the  error  in  the  com- 
puted values  of  W,  which  are  obtained  by  subtracting  the  second  column 
from  the  third;  and  the  fifth,  the  percentage  of  error.  The  percentage  of 
error  is  found  by  dividing  the  error  by  the  observed  value  of  W\  Thus 
0.2/3.1  =0.064,  or  6.4%. 


w 

10 

W 

0.29Tf 

Error 

%of 
error 

3.1 

2.9 

-0.2 

-6.4 

20 

5.8 

5.8 

0.0 

0 

30 

8.1 

8.7 

+  0.6 

+  7.4 

40 

11.2 

11.6 

+  0.4 

+  3.5 

I 


Note  the  part  played  by  mathematics  in  this  illustration 
the  scientific  method.  The  given  data  are  obtained  by  ohservi 
tion.  The  principles  of  graphic  representation  enable  us  t 
put  the  data  in  a  form  which  makes  reasonable  the  hypothesis 
that  the  graph  of  the  law  under  investigation  is  a  straight  line, 
and  that  the  law  is  represented  by  a  linear  equation.  By  d&- 
ductive  processes  we  determine  the  numerical  values  of  the 
coefficients  of  the  equation,  and  the  accuracy  of  the  represen- 
tation of  the  given  data  by  the  equation  found.  The  verification 


LINEAR  FUNCTIONS  81 

would  consist,  in  part,  of  repeating  the  experiment  a  number 
of  times,  varying  the  values  of  Wy  and  seeing  if  the  law  re- 
mained approximately  the  same.  But  a  more  satisfactory 
verification  would  be  to  deduce  from  this  law  some  other  law 
which  could  be  verified  by  an  experiment  of  a  different  sort. 
This  will  be  done  in  a  later  section. 

The  law  obtained  in  Example  1  can  be  stated  in  a  more  con- 
venient form.  The  force  tending  to  make  the  block  slide  is 
equal  to  W,  because  the  tension  of  the  string  is  the  same  at 
all  points.  This  force  acts  in  the  direction  of  the  surfaces  in 
contact,  and  when  motion  is  about  to  begin,  it  is  numerically 
equal  to  the  force  of  friction  which  is  preventing  the  block 
from  moving.  W  is  the  pressure  on  the  board  acting  perpen- 
dicularly to  the  surfaces  in  contact.  Hence  the  result  obtained 
may  be  stated  as  follows,  using  the  language  of  Section  21 : 

The  friction  of  wood  on  wood  varies  as  the  pressure  perpen- 
dicular to  the  surfaces  in  contact. 

If  different  kinds  of  wood,  or  other  substances,  be  used  in 
the  experiment,  the  value  of  m  obtained  would  not  be  the 
same.  But  extensive  experiments  have  shown  that  we  are 
reasonably  justified  in  stating  the  law: 

When  motion  is  about  to  take  place,  the  friction  between  two 
surfaces  varies  as  the  pressure  perpendicular  to  the  surfaces. 

Hence  if  F  denotes  the  force  of  friction  and  P  the  perpen- 
dicular pressure 

F  =  mP. 

The  constant  m  is  called  the  coefficient  of  friction,  and  it  is 
equal  to  the  ratio  of  the  friction  to  the  pressure  perpendicular 
to  the  surfaces  in  contact.  The  coefficient  of  friction  for  the 
block  and  board  in  Example  1  is  0.29. 

Example  2.  In  an  experiment  with  a  block  and  tackle,  the  pull  P 
necessary  to  raise  a  weight  W,  both  measured  in  pounds,  was  found  for 

W  HOP,  200.  300.  400,  1^"^  "^  *^  ^'"/^  "[^  *°  '^'  t^"^'^" 
-p-— ^y^ --^ -^ -— p     Fmd  approximately  the  equation  ex- 

'  '  '  '     pressing  P  as  a  function  of  W. 

Plotting  the  points  representing  the  pairs  of  numbers  in  the  table, 

using  values  of  W  as  abscissas,  it  is  seen  that  they  he  very  nearly  on  a 


82 


ELEMENTARY  FUNCTIONS 


straight  line.     This  line  ought  not  to  pass  through  the  origin,  because  if ' 
W  =  0,  B,  certain  force  P  is  necessary  to  raise  the  lower  block.     Hence  P 
is  approximately  a  linear  function  of  W  of  the  form 

P=.mW  +  b. 

A  good  value  of  m  is  the  average  of  the  slopes  of  the  lines  determined  1 

by  each  pair  of  points. 

Denoting  the  points  by  A,  B,  C,  D  respectively,  the  slope  of  the  line  | 

ABis 

59  -  37 


m 


200  -  100 


=  0.22. 


In  like  manner  we  find  the  value  of  m  for  each  of  the  lines  determined 
by  two  of  the  points  A,  B,  C,  D.  These  values  are  given  in  the  table 
following. 

Line     |    AB,        AC,        AD,        BC,        BD,         CD 


0.22,       0.25,       0.25,       0.28, 


0.24 


400  jy 


0.26, 

The  average  value  of 
m  is  found  to  be  m  = 
0.25.  Hence  the  func- 
tion required  has  the 
form,  approximately, 
P  =  0.25W  +  b.  (2) 
n  the  graph  of  (2) 
passes  through  the 
point  A,  the  coordi- 
nates of  A  must  satisfy 
(2),  and  hence 

37  =  0.25  X  100  +  b, 
whence 
6  =  12. 
In  like  manner,  we  find  the  value  of  b  on  the  assumption  that  each  of 
the  points  B,  C,  D,  lies  on  the  graph  of  (2).    The  results  are  given  in  the 
table. 

Point    I     A,        B,        C,        D 
b      I     12,        9,        12,       11 

The  average  of  these  values  of  6  is  6  =  11.  Substituting  this  value  in 
(2),  the  function  required  is,  approximately, 

P  =  0.25Tr  +  11. 

The  accuracy  with  which  this  equation  represents  the  observed  data  is 
shown  by  the  table  below.  The  first  two  columns  give  the  observed 
values  of  W  and  P.    The  third  column  gives  the  values  of  P  computed 


Fig.  46. 


LINEAR   FUNCTIONS 


83 


from  the  observed  values  of  W  by  means  of  (3).  The  fourth  gives  the  error 
in  the  computed  value  of  P  as  compared  with  the  observed  value,  while 
the  last  column  gives  the  percentage  of  error,  which  is  obtained  from  the 
second  and  fourth  columns. 


w 

Observed 
P 

Computed 
P 

Error 

Per  cent 
of  error 

100 

37 

36 

-  1 

-2.7 

200 

59 

61 

+  2 

+  3.3 

300 

87 

86 

-  1 

-1.1 

400 

111 

111 

0 

0 

property  of  a  linear  function  is  illustrated  in  the  table,  which  gives 


X 

X 

0 

y 

3 

1 

1 

5 

1 

3 

9 

1 

4 

11 

Ay 


values  of  x,  y,  Ax,  and  Ay  for  the  function  y  =  2x  +S. 
The  values  of  x  being  such  that  the  successive  values 
of  Ax  are  equal,  it  appears  that  the  successive  values 
of  Ay  are  also  equal.  That  this  is  always  the  case 
follows  from  the  fact  that  the  value  of  Ay /Ax  is  al- 
ways the  same  for  points  on  a  straight  line. 

Now  suppose,  as  in  the  examples  above,  the  points 
representing  a  given  table  of  values  appear  to  He  on 
a  straight  line.  To  test  the  accuracy  of  this  assumpn 
tion,  find  the  successive  values  of  Ax  and  Ay.  If  the  values  of  Ax  are 
equal,  and  if  those  of  Ay  are  nearly  equal,  we  are  justified  in  believing  that 
y  is  indeed  a  linear  function  of  x. 

The  labor  involved  in  computing  an  average  may  be  lessened  by  means 
of  the  following  rule:  Assume  a  number  x  which  appears  to  be  a  reason- 
able "guess  "  for  the  average  desired.  Subtract  x  from  each  of  the  given 
numbers,  find  the  average  of  the  differences,  and  add  it  to  x,  paying  due 
regard  to  signs  throughout. 

The  process  is  illustrated  for  the  average  of  the 
numbers  in  the  first  column,  the  value  of  x  being 
chosen  as  85.  The  differences  obtained  by  sub- 
tracting 85  from  each  of  the  numbers  are  given  in 
the  second  and  third  columns.  Their,  average  is 
obtained  by  adding  them  and  dividing  by  6,  the 
number  of  numbers  to  be  averaged,  and  is  found  to 
be  -  2.5.  This  must  be  added  to  85,  which  gives 
82.5  as  the  average. 

The  advantage  of  the  method  lies  in  the  fact  that 
it  involves  only  relatively  small  numbers,  which 
decreases  the  probability  of  error,  and  also  enables 
one  to  handle  simple  cases  mentally. 


85 

0 

70 

-15 

80 

-    5 

90 

+     5 

75 

-10 

95 

+     10 

+    15 

-30 

-   30 

6| 

-    15 

-   2.5 

85 

+  (2.5)  = 

=  82.5 

84  ELEMENTARY  FUNCTIONS 

The  proof  of  the  correctness  of  the  rule,  for  4  numbers  a,  6,  c,  d,  follows 
from  the  fact,  which  is  readily  verified,  that  the  equation 

a  +  b  +  c  +  d  _         (a  -x)  +  (b  -x)  +  (c  -  x)  +  (d  -  x) 

4 

is  true  no  matter  what  the  value  of  x  may  be. 


EXERCISES 

1.  In  an  experiment  on  friction  of  metal  on  metal,  values  of  W  and  W 
(see  Example  1)  were  found  as  in  the  following  table.  Find  the  equation 
connecting  W  and  W.    What  is  the  value  of  the  coefficient  of  friction? 

W      I     10,  20,  30,  40, 

W     I     1.9,         4.2,         6.1,        7.8, 

2.  In  an  experiment  on  the  friction  of  leather  on  metal,  values  of  W 
and  W  (see  Example  1)  were  determined  as  in  the  table.  Find  the  equa- 
W  I  10,  20,  30,  40,  tion  connecting  W  and  W.  What  is 
TF'  I  6.2,     11.7,     17.9,    24.3,     the  coefficient  of  friction? 

3.  The  table  gives  corresponding  values  of  P  and  W  (see  Example  2) 
PT  I    1 00       200         ^00        400       obtained  in  an  experiment  on  a  block 

-^-    .-,  o — ^Fr~i — iA>7  a — 1^1   I     and  tackle.     Find  the  relation  be- 
P     41.2,     74.6,     107.6,    141.5,     ,  d      j  w 

'    tween  P  and  W. 

4.  In  an  experiment  on  a  system  of  pulleys  in  which  all  but  one  were 

Tfl200,      400,      600,     1000,     movable   the  values  of  Tf  and  Pin 
'  —        —         —  the    table    were    found.     Fmd     the 


P     37.3,     61.8,    87.1,    137.2,  :.        ,  ^^     ,  J — 

'     equation  of  the  relation. 

5.  The  table  gives  the  length,  in  inches,  of  an  iron  rod  at  different  tem- 
n    0  10  20  30  40  Peratures,    in    degrees 

r    40,     40.0(;46,     40.0lk     40.ol45     40.0193,     ^^^f^^f^'     ^md  ^he 
'  '  length  I   at  any  tem- 

perature t.     (Assume  that  the  length  when  <  =  0°  is  exact.) 

6.  The  table  gives  the  length  in  centimeters  of  a  brass  rod  at  various 

temperatures  in  degrees  Centi- 

r-kTT^rb — ^^^^TTi na  Aa    o  a  aUa      g^ade.     Find  the  length  of  the 

M  36.405,     36.411,     36.416,3  6.424,         .     .   no  r.     .-       j  j  +u 

'  ,  '  '  '    rod  at  0    Centigrade,  and  the 

rate  of  change  of  the  length  of  the  rod  with  respect  to  the  temperature. 

7.  Find  the  volume  of  alcohol  at  any  temperature  from  the  table* 

t     I        10,  20,  40,  60, 

V    I     101.1,         101.9,         103.9,        106.0, 

8.  Determine  the  density  (see  page  61)  of  brass  from  the  table,  which 
5,         10,  15,         20,         gives  the  mass  m  of  pieces  of  brass 


m  I  41.2,    83.6,     124.1,    166.8,       with  the  volume  e;. 


LINEAR   FUNCTIONS  85 

9.   In  order  to  graduate  a  spring  balance,  the  extension  e  in  inches 

was  measured  for  different  weights  w^ 

w  I      2,  5y  7, 10^     as  in  the  table.     Find  to   hundredths 

T"!  0.52,     1.31,     1.84,    2.62,     of  an  inch  the  extension  due  to  one 

pound. 

10.  The  pressure  p,  per  square  foot,  at  various  depths  d,  in  feet,  under 

water  is  given  in  the  table.     Find  the 
d  I    20,        40,         60,        80,       pressure  at  a  depth  of  one  foot,  and  the 
p~|  1240,     2500,     3760,    4980,     rate  of  change  of  the  pressure  with  re- 
spect to  the  depth. 

11.  Barometer  readings  h  were  made  simultaneously  at  various  heights 

6  I  29.00,     28.88,     28.79,    28.45,    ^^'   |f /Z^*'   ^^"7".  ^  ,^^7  P^^^.*' 

^ — j^ Qg-^ 2or"^ — 495~^    ^^^^  *^®  ^^^  ^^"^^^  *^®  change  m 

'  '  '  '      altitude  as  a  function  of  the  change 

in  barometer  for  small  changes  in  altitude. 


MISCELLANEOUS  EXERCISES 

1.  Construct  the  line  through  the  point  (3,  2)  whose  slope  is  -  \,  and 
find  its  equation. 

2.  Construct  the  line  determined  by  the  points  (2,  1)  and  (4,  5),  and 
find  its  equation. 

3.  Construct  the  graph  of  the  equation  y  =  2t  +  Z.  Discuss  and  in- 
terpret the  graph  if  y  represents  the  distance  from  a  station  to  a  moving 
body  at  the  time  t;  if  y  represents  the  velocity  of  a  moving  body  at  the 
time  t. 

4.  The  volume  of  a  cake  of  ice  is  proportional  to  the  volume  of  the 
water  from  which  the  ice  was  obtained.  If  55  cubic  centimeters  of  ice 
are  made  by  freezing  50  cubic  centimeters  of  water,  how  much  water  will 
be  required  to  make  500  cubic  centimeters  of  ice?  Interpret  the  solution 
graphically. 

6.  A  train  is  moving  at  the  rate  of  35  miles  an  hour  when  the  brakes 

\  are  applied.  Six  seconds  later  the  train's  rate  is  17  miles  an  hour.  Con- 
struct the  graph  of  the  velocity  as  a  function  of  the  time,  and  find  its 

;  equation.     How  long  after  the  brakes  are  applied  will  the  train  come  to 

I  rest?    Check  the  result  graphically. 

i       6.   A  12-pound  specimen  of  copper  ore  lost  2.5  pounds  when  weighed  in 

I'  water.  How  much  copper  did  it  contain  if  8.9  pounds  of  copper  and  2.7 
pounds  of  the  material  with  which  the  copper  was  combined  each  lost  1 
pound  when  weighed  in  water? 

!  7.  How  many  cubic  centimeters  of  cork,  density  0.3,  must  be  combined 
with  75  cubic  centimeters  of  steel,  density  0.9,  in  order  that  the  combined 

I  mass  will  just  float  in  water?  Hint:  The  density  of  the  combination 
must  equal  that  of  water,  which  is  unity. 


86  ELEMENTARY  FUNCTIONS 

8.  Specific  gravity  is  the  ratio  of  the  density  of  a  body  to  that  of  another 
body  taken  as  a  standard.  Water  which  is  taken  as  the  standard  has  a 
density  of  unity  in  the  metric  system.  Hence  the  mass  of  a  substance  is 
equal  to  the  specific  gravity  times  the  volume. 

How  much  water  must  be  added  to  25  cubic  centimeters  of  concentrated 
hydrochloric  acid,  specific  gravity  1.20,  to  reduce  the  specific  gravity  to 
1.12? 

9.  How  much  water  must  be  added  to  30  cubic  centimeters  of  ammonia, 
specific  gravity  0.90,  to  raise  the  specific  gravity  to  0.96? 

10.  It  is  desired  to  reduce  the  specific  gravities  of  quantities  of  sul- 
phuric acid,  specific  gravity  1.84,  and  of  nitric  acid,  specific  gravity  1.42. 
How  much  water  must  be  added  to  80  cubic  centimeters  of  each  to  reduce 
the  specific  gravities  to  1.18  and  1.20  respectively? 

11.  What  mass  of  hydrogen  (molecular  weight  2)  will  be  displaced  by 
10  grams  of  zinc  (molecular  weight  65)  acted  upon  by  hydrochloric  acid 
if  the  amounts  exchanged  are  proportional  to  the  molecular  weights? 

12.  Find  the  money  value  of  pure  gold  in  a  $20  gold  piece,  if  one  ounce 
of  gold  is  worth  $20.66  and  the  coin  weighs  516  grains  and  is  ^^  pure. 

13.  A  tank  can  be  filled  by  one  pipe  in  3  hours,  and  emptied  by  a  second 
in  2  hours,  and  by  a  third  in  4  hours.  How  long  will  it  take  to  empty 
the  tank  when  it  is  full  if  all  the  pipes  are  opened? 

14.  The  report  of  a  gun  was  heard  in  3  seconds  at  a  place  3189  feet 
distant,  toward  which  the  wind  was  blowing;  and  in  2  seconds  at  a  place 
2074  feet  distant,  from  which  the  wind  was  blowing.  Find  the  velocity 
of  sound  and  of  the  wind. 

15.  A  railway  passenger  observed  that  a  train  moving  in  the  opposite 
direction  passes  him  in  2  seconds,  but  when  moving  in  the  same  direction 
it  passed  him  in  30  seconds.     Compare  the  rates  of  the  trains. 

16.  Two  cities  are  39  miles  apart.  If  A  leaves  one  city-two  hours  earlier 
than  B  leaves  the  other,  they  meet  two  and  a  half  hours  after  B  starts. 
Had  B  started  at  the  same  time  as  A,  they  would  have  met  three  hours 
after  they  started.     How  many  miles  an  hour  does  each  walk? 

17.  The  accuracy  with  which  one  can  weigh  an  object  on  a  balance  de- 
pends on  the  number  of  spaces  on  the  scale  which  the  pointer  moves 

when  a  small  weight  is  added.  The 
W  I      5,  25,        50.       100,     table  gives  the  deflection,  D,  of  the 

D    I    1.99,     1.72,     1.76,    0.95,     pointer  with  a  weight  W  in  the  pan 

when  one  milligram  is  added  to  W. 
Determine  D,  approximately,  as  a  function  of  W. 


CHAPTER  III 


ALGEBRAIC   FUNCTIONS 


28.  Introduction.  In  this  chapter  we  shall  study  the  proper- 
ties (Sections  10  to  13)  of  certain  types  of  algebraic  functions 
(Section  14). 

Among  these  types  are  the  quadratic  function,  ax^  +  bx  +  c, 
which  occurs  in  the  solution  of  quadratic  equations,  and  the 
function  ax^.  The  latter  is  an  integral  rational  function  if  n 
is  a  positive  integer,  a  rational  fractional  function  if  n  is  a 
negative  integer,  and  an  irrational  function  if  n  is  a  fraction. 
Other  types  are  the  linear  fractional  function,  {ax  -\-  h)/{cx  +  d), 
and  polynomials,  which  are  studied  in  order  to  obtain  a 
method  for  solving  equations  of  higher  degree  than  the  second, 
an  extension  of  the  solution  of  Unear  and  quadratic  equations. 

These  functions  find  frequent  appUcation  in  many  fields. 
For  example,  the  quadratic  function  appears 
in  the  theory  of  falling  bodies,  and  among 
the  laws  which  can  be  represented  by  a 
function  of  the  type  ax"  are  Newton's  law 
of  gravitation,  and  Boyle's  law  connecting 
the  pressure  and  volume  of  a  gas. 

In  the  study  of  quadratic  functions  we 
shall  proceed  from  special  cases  to  the 
general  case  by  methods  which  are  impor- 
tant in  other  connections. 

29.  Graph  of  jc^.    Since  (-  x)^  =  x^,  the 
graph  is  symmetrical  with  respect  to  the  y-axis 
(Theorem  lA,  page  23).     Hence  the  ^/-axis 
is  called  an  axis  of  symmetry. 
only  positive  values  of  x. 

87 


T   4      1~ 

H    .    ^ 

t          4 

I       i 

X       t 

^        1 

f     4 

X-^  t 

\]  / 

ik^  _i 

-'-'-'"  ' ' '" 

0, 


FiQ.  47. 

1>     2, 
1, 


0,     1,     4,     9, 
The  table  of  values  need  include 


88 


ELEMENTARY  FUNCTIONS 


It  is  readily  seen  that  the  intercepts  are  x  =  0  and  y  =  0, 
Hence  the  curve  passes  through  the  origin,  but  does  not  cut 
the  axes  elsewhere. 

Since  x^  is  positive  if  x  is  real,  no  part  of  the  curve  lies  helow 
the  X-axis. 

As  X  increases,  so  also  does  x^,  and  hence  the  curve  broadens 
out  as  it  rises.  If  x  >  1,  x^  increases  more  rapidly  than  x,  so 
that,  to  the  right  of  x  =  1,  the  curve  rises  more  rapidly  than  it 
broadens  out.    The  origin  is  a  minimum  point. 

30.  Graphs  of  ax^  and  af(x).  Consider  the  graph  of  the 
function  2x\  The  ordinate  of  any  point  on  the  graph  is  twice 
that  of  the  point  with  the  same  ab- 
scissa on  the  graph  of  x^.  Hence  the 
graph  of  2x^  may  be  plotted  by  doubling 
the  ordinates  of  a  number  of  points  on 
the  graph  of  x^,  and  drawing  a  smooth 
curve  through  the  points  so  obtained. 
In  like  manner,  the  graph  of  Sx^  may 
be  obtained  by  trebling  the  ordinates 
of  points  on  the  graph  of  x^,  of  Jx^  by 
bisecting  them,  etc. 

The  ordinates  of  points  on  the  graph 

of  —  2x'^  are  numerically  twice  those  of 

points  on  the  graph  of  x^,  but  as  they  are 

negative,  the  points  on  the  graph  lie 

below  the  x-axis.    The  graphs  of  2x'^  and  -  2x^  are  symmetrical 

to  each  other  with  respect  to  the  x-axis.  i 

From  the  method  of  constructing  these  graphs  we  see  that 

The  y-axis  is  an  axis  of  symmetry  of  the  graph  of  ax^.     The 

curve  runs  up  or  down,  and  the  origin  is  a  minimum  or  maximum 

point,  according  as  a  is  positive  or  negative. 

If  a  is  positive,  the  graph  of  ax^  rises  more  or  less  rapidly  than 
that  of  x^,  for  which  a  =  1,  according  as  a  is  greater  or  less  than 
unity.  The  larger  the  value  of  a,  the  more  rapidly  the  curve 
rises,  and  the  smaller  the  value  of  a  is,  the  less  rapidly  it  rises. 
Any  one  of  these  curves  is  called  a  parabola.  The  axis  of 
symmetry  of  a  parabola  is  called  its  axis. 


TI-;    in 

-W-W 

^4t_    Ifr 

^u     Cy 

_^B      /// 

Sa,*   /// 

\^>f 

^-'-^ 

771^  SV 

ttt-l  m 

i/lrHJ 

rit '  iffi 

Fig.  48. 


ALGEBRAIC  FUNCTIONS 


89 


I 

^KThe  reasoning  employed  above  may  be  used  to  prove  the 

^^Theorem.     The  graph  of  af{x)  may  be  obtained  by  multiplying 

by  a  the  ordinates  of  points  on  the  graph  of  f{x).    Corresponding 

points  on  the  two  graphs  lie  on  the  same  or  opposite  sides  of  the 

X-axis  according  as  a  is  positive  or  negative. 

If  a  =  -  1  the  graph  is  symmetrical  to  that  of  f(x)  with  re- 
spect to  the  a;-axis. 

This  theorem  is  the  second  one  we  have  considered  belonging 
to  the  set  of  relations  between  pairs  of  functions  and  their 
graphs  (see  the  last  paragraph  but  one  on  page  43). 

31.  Translation  of  the  Coordinate  Axes.  Consider  a  system 
of  coordinate  axes  with  the  origin  O,  and  a  second  system, 
parallel  to  the  first,  with  origin  O'.  Replacing  the  first  system 
by  the  second  is  called  translating  the  axes.  By  the  equations 
for    translating    the    axes   we 


^' 


Pry 


mean  equations  which  express 
the  coordinates  of  a  point  re- 
ferred to  the  first,  or  old,  axes 
in  terms  of  the  coordinates  of 
the  same  point  referred  to  the 
second,  or  new,  axes. 

Let  the  old  and  new  coordi- 
nates of  any  point  P  be  respec- 
tively {x,  y)  and  {x\  y'),  and 
let  the  old  coordinates  of  the 
new  origin,  0',  be  Qi,  k).  From 
the   figure  we  readily  obtain 

Theorem  1.     The  equations  for  translating  the  axes  are 


V' 


Ei 


X' 


Fig.  49. 


X  =  x'  -\-hf\ 

y  =  y'  +  hi 


(1) 


1^  where  Qi,  k)  are  the  coordinxites  of  the  new  origin. 

Suppose  that  the  graph  of  an  equation  in  x  and  y  has  been 
plotted.  To  find  the  equation  of  this  graph  referred  to  new 
axes  parallel  to  the  old  axes  we  substitute  in  the  given  equation 
the  values  of  x  and  y  given  by  (1).  The  following  examples 
illustrate  the  utility  of  the  theorem. 


90  ELEMENTARY  FUNCTIONS 

Example  1.     Given  the  equation 

2/  =  x2  -  6x  +  13,  (2) 

translate  the  axes  so  that  the  new  origin  is  the  point  (3,  4)  and  find  the 
equation  in  the  new  coordinates  which  has  the  same  graph  as  (2).     Plot 
the  graph,  and  state  its  most  noteworthy  properties. 
By  Theorem  1  we  have 


M     ' 

; 

^^ 

7 

t 

t 

3 

' 

k 

iZ     ^ 

'0 

T      ^    ei' 

0 

-^ 

a;  =  x'  +  3, 

2/  =  ?/'  +  4. 

Substituting  these  values  in  (2),  we  have 

y'  +  4.=  {x'  +  3)2  -  6(x'  +  3)  +  13. 

Removing  the  parentheses, 

y'  +  4  =  x'2  +  6x'  +  9  -  6x'  -  18  +  13, 
or 


(3) 


(4) 


(5) 


Fig.  50.  The  graph  of  (5),  plotted  on  the  new  axes,  is  the 

same  as  the  graph  of  (2),  referred  to  the  old  axes. 
But  the  graph  of  (5)  is  a  parabola  which  is  easily  plotted  on  the  new  axes. 
It  is  shown  in  the  figure.  This  curve  is  also  the  graph  of  (2)  when  plotted 
on  the  old  axes.  From  it  we  see  that  the  axis  of  symmetry  of  the  parab- 
ola, the  2/'-axis,  is  the  line  x  =  3,  and  that  the  minimum  point  is  the 
new  origin  (3,  4). 

In  this  example  it  turned  out  that  the  equation  obtained  by 
translating  the  axes  was  much  simpler  than  the  given  equation. 
The  question  arises :  If  an  equation  is  given,  how  can  we  de- 
termine how  to  move  the  axes  so  as  to  obtain  a  simpler 
equation  with  the  same  graph?  The  method  of  doing  this  is 
illustrated  in 

Example  2.    Simplify  the  equation 

2/  =  2x2  -  &c  +  11  (6) 

by  translating  the  axes,  and  construct  the  graph. 

First  solution.     Substituting  in  (6)  the  values  of  x  and  y  given  by  (1), 

we  get 

?/'  +  fc  «  2{x'  +  A)2  -  8(x'  +  /i)  +  11 

=  2x'2  +  4/ix'  H-  2/i2  _  8x'  -  8^1  -h  11, 
whence  y'  =  2x"'  +  {Ah  -  8)x'  +  2/i2  -  8/i  +  11  -  fc.  (7)'i 

This  equation  will  be  very  simple  if  the  coeflficient  of  x'  and  the  constant 
term  are  zero,  that  is,  if 

4A-8  =  0.|  (gy 


I 


and 


2^2  -  8A  +  11  -  fc 


y.\ 

^J 

' 

^ 

1 

J 

'K 

g    3     i 

s    .^J 

0 

X 

ALGEBRAIC  FUNCTIONS  91 

Solving  these  equations  for  h  and  k,  we  get 

h  =  2,  \  /qx 

,and  /c  =  2x22-8x2  +  ll  =  3./.  ^""^ 

If  tiiese  values  are  substituted  in  (7)  we  obtain 

y'  =  2x'\  (10) 

Hence,  if  the  axes  be  translated  so  that  the 
origin  is  moved  to  the  point  v2,  3),  equation  (6) 
assumes  the  simpler  form  (10).  Plotting  the 
graph  of  (10)  on  the  new  axes  we  get  the  curve  in 
the  figure. 

Second  solution.     The  form  of  the  given  equa-  ^iQ-  SI. 

tion  (6)  may  be  changed  as  follows: 

2/  =  2x2  -  8x  +  11 
=  2(x2  -  4a;)  +  11 
=  2(x2  -  4a;  +  4  -  4)  +  11 
-  2(a;2  -  4x  +  4)  -  8  +  11 
=  2(x  -  2)2  +  3, 
whence  y  -Z  =  2{x  -  2)^. 

It  is  seen  by  inspection  of  this  equation  that  if  we  set 

X  =  x'  +  2,  "I 
and  y  =  y'  +  S,  ) 

we  will  obtain  the  simpler  equation 

y'  =  2x'2, 

which  is  identical  with  (10).  Equations  (11;  are  the  equations  for 
translating  the  axes  so  that  the  origin  is  moved  to  the  point  (2,  3)  (by 
Theorem  1). 

Let  the  graph  of  any  function  y  =  f{x)  be  given.     In  order 
to  move  the  x-axis  up  k  units,  leaving  the  2/-axis  unchanged, 
k  we  set  y  =  y'  +k,  obtaining  y'  -\-k  =  f{x),  or 

■  y'=f(x)-k.  (12) 

;  The  graph  of  (12)  is  identical  with  the  given  graph,  if  the 

j  equation  is  plotted  on  the  new  axes. 

Now  suppose  that  the  graph  of  (12)  is  plotted  on  the  original 
axes.    It  may  be  obtained  by  moving  the  given  graph  down  k 

I  units  (see  the  Theorem  in  Exercise  3,  page  19).     Hence  the 

I  graph  of  (12)  may  be  interpreted  in  two  ways,  which  are  es- 
sentially identical  since  the  effect  of  moving  the  a;-axis  up  k 


(11) 


92 


ELEMENTARY  FUNCTIONS 


units,  and  erasing  the  original  x-axis,  is  the  same  as  moving  the 
graph  down  k  units  and  erasing  the  original  curve. 

In  Uke  manner,  if  we  set  x  =  x'  +  h,  and  leave  y  imchanged, 
we  get 

y=f{x'  +  h),  (13) 

The  graph  of  this  equation  is  identical  with  the  graph  of  /(x), 
if  plotted  on  the  original  x-axis  and  a  new  ?/-axis  h  units  to  the 
right  of  the  old.  But  the  effect  of  moving  the  2/-axis  h  units  to 
the  right  and  erasing  the  old  2/-axis  gives  the  same  figure  as 
moving  the  curve  h  units  to  the  left  and  erasing  the  old  curve. 
Hence  the  graph  of  (13)  referred  to  the  original  axes  may  be  ob- 
tained by  moving  the  graph  of  f{x)  h  units  to  the  left. 


A  B 

Fig.  62. 

In  plotting  (13)  on  the  original  axes  it  is  convenient  to  write 
X  in  place  of  x'.    We  thus  obtain 

Theorem  2.  The  graph  of  f{x  +  h)  may  he  obtained  by  moving 
the  graph  of  f(x)  h  units  to  the  left.  The  motion  will  be  to  the 
right  if  h  is  negative. 

This  theorem,  for  which  we  find  application  in  later  chap- 
ters, should  be  associated  with  the  last  paragraph  but  one  on 

page  43. 

EXERCISES 

1.  Plot  on  the  same  axes  the  graphs  of: 

(a)  x2,  3x2,  |2;2,   -  ^x\ 

(b)  x^  -  4x,  2(x2  -  4x),  Ux^  -  4x). 

(c)  x^  +  3,  2(x2  +  3),  Ux^  +  S). 

(d)  (X  -  2)2,  Six  -  2)2,  lix  -  2)2. 

2.  Find  the  value  of  o  if  the  graph  of  y  -  ox*  passes  through  the  point 
(2,  3),  and  construct  the  graph. 

1 


ALGEBRAIC  FUNCTIONS  93 

I        3.  Show  that  the  points  (2, 1),  (3, 2J),  and  (4,  4)  lie  on  one  of  the  parab- 

!  olas  ax^. 

1        4.   Show  that  one  of    the    curves   ax^    passes    through    any    point 

\  Pi(xi,  2/i). 

[        6.   Find  the  average  rate  of  change  of  y  with  respect  to  x,  for  the  inter- 

I  vals  from  x  =  0  to  x  =  1,  from  a;  =  1  to  a;  =  2,  from  a;  =  2  to  a;  =  3,  and  for 

I  any  interval  if  (a)  y  =  x^\   (b)  2/  =  2a;2;   (c)  y  =  ^x^;   (d)  y  =  ax^. 

6.  Translate  the  axes  to  the  new  origin  indicated,  and  construct  both 
pairs  of  axes  and  the  graph: 

(a)  y  =  x^  +  S,  O'(0,  3).  (b)  y  =  x^  -  4x,  0'(2,  -  4). 

(c)  2/  =  -  x^  0'(2,  -  4).  (d)  y  =  1/x,  0'(3,  2). 

7.  Using  Theorem  2,  Section  31,  construct  on  the  same  axes  the  graphs 
of  x\  (x  +  2)2,  and  (x  -  3)2. 

8.  Construct  the  graphs  of  x^  and  x^  +  8x  +  16  on  the  same  axes. 

9.  On  the  same  axes  construct  the  graphs  oi  y  =  x^,  y  =  2x2,  y  =  (x  +  2)^, 
y  =  a;2  +  2. 

10.  Simplify  the  following   equations   by   translating  the   axes.     In 
each  case,  construct  both  pairs  of  axes  and  the  graph.     Determine  the 
!  axis  of  symmetry  and  the  maximum  or  minimum  point  in  the  given  co- 
ordinates. 

(a)  x2  +  6x  +  4.  (b)  4x  -  x\ 

(c)  2x2  _  16a;  ^  35.  (d)  Sx^  -  12x  +  5. 

(e)   -  x2  +  2x  +  5.  (f)    -  2x2  _  i2x  -  20. 

32.  Instantaneous  Velocity.    If  a  ball  is  dropped,  its  velocity 

!  changes  continually.    An  approximate  value  of  what  we  mean 

when  we  speak  of  its  velocity  at  some  given  instant  is  given  by 

I  the  average  velocity  in  an  interval  of  time  A^  beginning  at 

I' that  instant.     The  smaller  the  value  of  A^  is,  the  more  ac- 

■  curate  is  the  approximation.    A  precise  notion  of  the  velocity 

■  at  an  instant  is  given  by  the 

Definition.  The  velocity  of  a  body  at  an  instant,  or  its  in- 
;  stantaneous  velocity,  is  the  limit  *  of  the  average  velocity  in  an 
i  interval  At  beginning  at  that  instant,  as  the  interval  At  ai> 
j  preaches  zero. 

The  computation  of  an  instantaneous  velocity  is  illustrated 
in  the  following  example. 

*  The  limit  of  a  variable  is  a  constant  such  that  the  numerical  value  of 
i  the  difference  between  the  variable  and  the  constant  becomes  and  r&- 
j  mains  less  than  any  assigned  positive  number,  however  small. 


94  ELEMENTARY  FUNCTIONS 

Example.  If  a  ball  is  dropped,  its  distance  from  the  starting  point  at 
any  time  <,  in  seconds,  is  given  by 

s  =  lQt\  (1) 

Find  the  velocity  at  any  time. 

The  distance  s  at  any  time  t  is  given  by  (1).  If  the  distance  changes  by 
As  in  the  next  A^  seconds,  the  distance  at  the  time  t  +  At  will  be  s  +  As. 
These  corresponding  values  of  the  distance  and  time  must  satisfy  (1), 
and  hence 

s  +  As  =  16(i  +  At)^  =  16f2  +  32f  At  +  IQAtK    (2) 

Subtracting  (1)  from  (2)  we  obtain 

As  =  S2t  At  +  lQAt\ 

As 
and  dividing  by  At,  ^  =  32<  +  IQAt.  (3) 

This  is  the  average  velocity  during  an  interval  of  At  seconds  beginning 
at  the  time  t.  By  the  definition  above,  the  velocity  at  the  time  t  is  the 
limit  of  (3)  as  At  approaches  zero.     Denoting  it  by  y,  we  get 

v^S2t,  (4) 

since  IQAt  approaches  zero  when  At  approaches  zero. 

33.  Rate  of  Change.  Slope  of  Tangent  Line.  A  generaliza- 
tion of  the  idea  of  instantaneous  velocity,  which  may  be  de- 
scribed as  the  rate  of  change  of  the  distance  s  with  respect  to 
the  time  t,  is  given  in  the 

DEFmiTiON.  The  rate  of  change  of  y  with  respect  to  x  for  a 
given  value  of  x  is  the  limit,  as  Ax  approaches  zero,  of  the 
average  rate  of  change  of  y  with  respect  to  x  in  the  interval  froi 
X  to  X  +  Ax. 

If  1/  is  a  function  of  x, 

y-Ax),  (1] 

the  rate  of  change  of  y  with  respect  to  x  may  be  determined 
follows : 

Substitute  x  +  Ax  for  x  and  y  +  Ay  for  y  in  (1).    The  result, 

2/  +  A2/=/(x  +  Ax),  (2) 

is  true  because  Ay  is  the  change  in  y  due  to  a  change  of  Ax  in 
X,  and  hence  x  +  Ax  and  y  -{-  Ay  are  corresponding  values  oi 
the  independent  variable  and  the  function. 


ALGEBRAIC  FUNCTIONS 


95 


Fig.  53. 


Subtract  (1)  from  (2),  which  gives  Ay,  and  divide  by  Ax,  which 
gives  Ay /Ax,  the  average  rate  of  change  of  y  in  the  interval 
Ax.  Then  find  the  limit  of  the  average 
rate  of  change  as  Ax  approaches  zero. 

To  interpret  this  process  graphi- 
cally we  shall  need  the 

Definition.  If  P  is  a  given 
point  on  a  curve  and  Q  any  other 
point  on  it,  the  line  tangent  to  the 
curve  at  P  is  the  limiting  position  of 
the  secant  PQ  as  Q  moves  along  the 
curve  and  approaches  P. 

The  points  P(x,  y)  and  Q{x  -\-  Ax,  y  +  Ay)  are  on  the  graph 
of  (1),  and  Ay /Ax  is  the  slope  of  the  secant  PQ.  As  Ax  ap- 
proaches zero,  Ay /Ax  approaches  the  rate  of  change  of  y;  also, 
Q  approaches  P,  the  secant  PQ  approaches  the  tangent  PT, 
and  the  slope  of  PQ  approaches  that  of  PT.    Hence, 

The  rate  of  change  of  y  with  respect  to  x  is  represented  by  the 
dope  of  a  line  tangent  to  the  graph  of  y. 

The  tangent  to  the  graph  of  2/  at  a  maximum  or  minimum 
point  is  horizontal,  and  hence  its  slope  is  zero,  and  hence,  also 
the  rate  of  change  of  y. 

This  fact  affords  a  general  method  of  finding  maximum  and 
minimum  points. 

The  slope  of  the  tangent  line  and  the  rate  of  change  of  the 
function  should  be  added  to  the  table  of  properties  of  graphs 
and  functions  on  page  42. 

Example  1.    Find  the  rate  of  change  of 

y  =  ax\  (3) 

and  discuss  its  meaning  for  the  graph. 
Replacing  x  and  y  by  x  +  Ax  and  y  +  Ay  respectively,  we  get 

y  +  Ay  =  aix  +  Ax)^  =  ax^  +  2ax  Ax  +  a  Ax^. 

btracting  (3)  from  (4), 

Ay  =  2ttx  Ax  +  aAx\ 
Ay 


(4) 


Hviding  by  Aa;, 


Ax 


=  2ax  +  a  Ax, 


(5) 


ich  is  the  average  rate  of  change  of  y  in  any  interval  Ax. 


96 


ELEMENTARY  FUNCTIONS 


Letting  Ax  approach  zero,  the  limit  of  (5)  is 

m  =  2ax, 


(6) 


the  required  rate  of  change,  or  the  slope  of  the  tangent  line  at  any 
point. 

If  X  =  0,  we  have  m  =  0,  so  that  the  parabola  which  is  the  graph  of  (3) 
is  tangent  to  the  x-axis  at  the  origin. 

For  a  given  value  of  x,  the  larger  the  value  of  a,  the  larger  also  is  the  slope 
of  the  tangent  line.  This  justifies  the  statement  made  in  Section  80  that 
"  the  larger  the  value  of  a  the  more  rapidly  the  curve  rises." 

If  a  =  1,  so  that  y  =  x^,  we  have  m  =  2x.  Hence  the  slope  of  the  tangent 
line  is  less  or  greater  than  1  according  as  x  is  less  or  greater  than  |.  There- 
fore, to  the  right  of  x  =  ^,  the  curve  rises  more  rapidly  than  it  broadens 
out.    (Compare  with  the  statement  in  Section  29). 

For  any  given  value  of  a,  the  larger  the  value  of  x  the  larger  is  the 
value  of  m.  Hence  the  curve  rises  more  and  more  rapidly  as  it  runs  to 
the  right. 

Example  2.  Find  the  slope  of  the  line  tangent  at  any  point  to  the 
graph  of 


y  =  x^  -  4x. 


(7) 


Find  the  slope  at  the  point  for  which  x  =  3,  construct  the  line,  and  find  its 
equation.    Find  also  the  coordinates  of  the  maximum  or  minimum  points. 

The  table  of  values  and  the  graph  are 
readily  constructed. 

Replacing  x  by  x  +  Ax  and  y  by  ?/  +  Ay 
in  (7)  we  get 

2/  =  (x  +  Ax)2  -  4(x  +  Ax) 
or     ?/  =  x2  +  2x  Ax  +  Ax2  -  4x  -  4Ax.     (8) 

Subtracting  (7)  from  (8), 

At/  =  2x  Ax  +  Ax2  -  4Ax. 

Dividing  by  Ax, 

Ax 


l^i       T 

k  it  ^ 

t            4 

V            t^ 

A          tt 

J      'k     - 

■10        19     3        /  5     6      7    b  X 

1   ,                          f 

X        J 

.3     -L 

.    S2    " 

^x 

Fig.  64. 


4. 


(9) 


Passing  to  the  limit  as  Ax  approaches  zero,  the  slope  of  the  tangent  line 


at  any  point  is 


jr  I  -  1,  0,       1,      2,      3,  4,  5, 
y  I  ^5,  0,  -  3,  -  4,  -  3,  0,  6, 


2x 


(10) 


At  the  point  A  (3,  -  3)  the  slope 
of  the  tangent  line  is  therefore  m 
=  2x3-4  =  2,  and  the  line  may  b« 
drawn  by  the  construction  on  page  5% 


ALGEBRAIC  FUNCTIONS  97 

Using  the  point-elope  form  of  the  equation  of  a  straight  Hne  (page  66), 
the  equation  of  the  line  tangent  at  A  is 

I  2/  +  3  =  2(x-3), 

or  2/  =  2x  -  9. 

As  a  check,  notice  that  the  intercept  on  the  ^-axis,  -  9,  agrees  with  the 
line  as  constructed. 

At  a  maximum  or  minimum  point  the  tangent  line  is  horizontal,  and 
'  hence  the  slope  m  given  by  (10)  must  be  zero.     Hence 

I  2x  -  4  =  0, 

'  whence  x  =  2. 

Substituting  this  value  in  (7),  the  minimum  value  of  y  is  2/  =  -  4.  Hence 
the  minimum  point  is  the  point  D(2,  -  4). 

EXERCISES 

1.  A  ball  is  rolled  down  an  inclined  plane.  Its  distance  from  the  start- 
I  ing  point  is  given  by  s  =  ^t^  +  4L  Find  the  velocity  at  any  time;  at  the 
I  instant  ^  =  3.     Plot  the  graphs  of  s  and  v  on  the  same  axes,  and  from  them 

describe  the  motion. 

2.  Prove  that  a  line  tangent  to  a  circle,  as  defined  in  Section  33,  is  per- 
pendicular to  the  radius  drawn  to  the  point  of  contact,  using  the  form  of 
reasoning  of  elementary  geometry. 

3.  The  distance  from  a  fixed  station  to  a  moving  body  is  given  by 
s  =  f2  _  3^^  Find  the  velocity  at  any  time,  and  show  that  the  acceleration 
is  constant.  Plot  the  graphs  of  s  and  v  on  the  same  axes,  and  describe  the 
motion  from  t  =  0  to  t  ="  5. 

4.  Find  the  slope  of  the  line  tangent  at  any  point  to  the  graph  of  each 
I  of  the  functions  below.  Find  the  slope  of  the  line  tangent  at  the  point  for 
I  which  X  =  2,  construct  the  line,  and  find  its  equation.  Find  the  coordi- 
\  nates  of  the  maximum  or  minimum  points. 

(a)  4  -  a;2.      (b)  y  =  3x  -  x\      (c)  y  =  x^  -  1.      (d)  x^  -4y  +  2x  =  0. 

i  6  Tabulate  the  values  of  the  slope  of  the  line  tangent  to  the  graph  of 
j  r*  at  the  points  for  which  a:  =  —  3,  —  2,  —  1,  0,  1,  2,  3.  What  can  be  said 
I  of  the  value  of  m  as  a;  increases?  Does  the  value  of  m,  the  slope  of  the 
;  tangent  line,  always  increase  as  x  increases  if  the  graph  is  concave  up- 
i  ward?     Is  the  converse  true? 

If  a  curve  is  concave  downward,  how  does  the  slope  of  the  tangent  line 
:  change  as  x  increases?     Is  the  converse  true? 

1  6.  Find  the  maximum  and  minimum  points  oi  y  =  x^  -  x,  and  then  plot 
[the  graph. 

i     7.   Find  the  equation  of  the  line  tangent  to  the  graph  of  x^  at  the  point 
for  which  x  =  1;   for  which  x  =  2;  for  which  x  =  4.     Find  the  intercept 


98  ELEMENTARY  FUNCTIONS 

of  each  line  on  the  y-axis.     How  does  the  intercept  compare  with  the 
ordinate  of  the  point  of  contact? 

8.  Prove  that  if  two  lines  are  perpendicular  the  slope  of  one  is  the 
negative  reciprocal  of  the  slope  of  the  other. 

Definition.  The  line  perpendicular  to  a  line  tangent  to  a  given  curve 
at  the  point  of  tangency  is  called  a  normal  to  the  curve. 

The  slope  of  the  normal  at  any  point  may  be  found  from  that  of  the 
tangent  by  the  preceding  exercise. 

9.  Find  the  equations  of  the  tangent  and  normal  to  the  graph  ol 
y  =  x^  4-  2x  at  the  point  (1,  3).     Construct  the  figure. 

10.  Find  the  equation  of  the  line  normal  to  the  graph  of  x^  at  the  poini 
for  which  x  =  1;   for  which  x  =  3;   for  which  x  =  5.     Find  the  intercept' 
of  each  line  on  the  2/-axis.     How  does  the  intercept  compare  with  the  or- 
dinate of  the  point  at  which  the  normal  cuts  the  curve? 

11.  Find  the  acceleration  of  a  moving  body  if  its  velocity  is  given  by 
V  =  2t^  -  5t.  Plot  the  graphs  of  v  and  o  on  the  same  axes  and  discuss  the 
variation  of  both  functions. 

12.  If  the  position  of  a  moving  body  is  given  by  the  equation  s  =  t^  -  % 
find  the  velocity  and  acceleration.  Plot  the  graphs  of  s,  v,  and  a  on  the 
same  axes. 

13.  Find  the  acceleration  of  a  body  if  its  position  is  given  by  s  =  4:(^  -  fi. 

14.  If  a  ball  is  dropped  the  velocity  after  the  ball  has  fallen  s  feet  is 
given  by  v^  =  2gs.    Find  the  rate  of  change  of  s  with  respect  to  v. 

34.  Graph  of  the  Quadratic  Function  ax^  +  bx  +  c.  Denote 
the  function  by  y  so  that 

y  =  ax^  +  hx  +  c.  (1) 

In  order  to  determine  the  form  of  the  graph  we  shall  first 
show  that  it  always  has  a  single  maximmn  or  minimum  point, 
and  then  translate  the  axes  so  as  to  have  this  point  for  the  new 
origin. 

To  find  the  maximum  or  minimum  point  we  need  first  the 
slope  of  the  tangent  line  at  any  point.  In  (1)  replace  x  by 
X  +  Ax  and  y  by  y  +  Ay,    This  gives 

y  +  Ay  =  a{x  +  Ax)^  +  h(x  +  Ax)  +  c, 
or 

y  +  Ay  ^  ax"^  +  2ax  Ax  +  Ax^  -\-hx  +  bAx  +  c.  (2) 

Subtracting  (1)  from  (2), 

Ay  ==2axAx-{-Ax^  +  b  Ax, 


J 


ALGEBRAIC  FUNCTIONS  99 

and  dividing  by  Ax, 

■jr  =  2ax  -f-  Ax  +  6. 

Ax 

Passing  to  the  limit  as  Ax  approaches  zero,  the  slope  of  the 
tangent  line  at  any  point  is 

m  =  2ax-\-  b.  (3) 

The  tangent  hne  will  be  horizontal  if  m  =  0,  that  is,  if 
2ax  +  b  =  0,  whence 

.=  -A.  (4) 

Substituting  this  value  in  (1)  the  corresponding  value  of  y  is 

(J)    4,ac  —  b'^\ 
-  — ,  — J j  is  either  a  maximum  or  a  mini- 
mum point.    To  translate  the  axes  so  that  O'  is  the  new  origin 
we  set 

,      b  ,      4:ac  -  b^ 

Substituting  these  values  in  (1), 

,     iac  -V        /  „      bx'      6n      ,  /  ,      6  \  , 

=  ax'2-6x'+f^  +  5x'-|+c, 

4a 
or  y'  =  cix'\  (6j 

The  graph  of  this  equation  is  a  parabola  (Section  30)  con- 
gruent to  the  graph  of  ax^.  Its  axis  of  symmetry  is  vertical, 
and  the  curve  runs  up  or  down  according  as  a  is  positive  or 
negative. 


100  ELEMENTARY  FUNCTIONS 

Then  since  the  graph  of  (1),  plotted  on  the  old  axes,  is  identi- 
cal with  that  of  (6),  plotted  on  the  new  axes,  we  have  the 

Theorem.  The  graph  of  the  quadratic  function  ax^  +  hx  -{-  c 
is  a  paraholaj  with  vertical  axis,  which  is  congruent  to  the  graph 
of  ax^.  It  runs  up  or  down  according  as  the  coefficient  of  x^  is 
positive  or  negative. 

Example.  If  a  ball  is  thrown  vertically  upward  with  a  velocity  of  80 
feet  per  second,  its  height  s,  in  feet,  after  t  seconds,  is  given  by  the  equation 

s  =  -  16^2  +  80t.  (7) 

Find  when  the  ball  will  be  highest,  how  high  it  will  rise,  and  compare 
the  time  of  rising  with  that  of  falling.     Construct  the  graph. 

At  the  highest  point  the  velocity  of  the  ball  is  zero,  and  we  therefore 
seek  first  the  velocity  at  any  time.  Replacing  thy  t  +  At  and  shy  s  +  As 
we  get 

s  +  As  =  -  m{t  +  Aty  +  80(<  +  At), 
or  s  +  As  =  -  16^2  _  32^  A«  -  IQAt^  +  80^  +  80A^  (8) 

Subtracting  (7)  from  (8), 

As  =  -  32t  At  -  16Ai2  +  SOAt, 
and  dividing  by  A^, 

^  =  -  32f  -  IQAt  +  80. 

Passing  to  the  limit  as  At  approaches  zero,  the  velocity  at  any  time  is 

v=  -S2t  +  80,  f  (9) 

since  the  limit  of  the  average  velocity  As/ At  is  the  instantaneous  velocity. 
At  the  highest  point  reached  by  the  ball  v  =  0,  and  hence  <  =  |f  =  | 
seconds.     Substituting  this  value  in  (7)  the  maximum  value  of  s  is 

or:  K 

s=-16x'=r  +  80x^=-100  +  200  =  100  feet. 
4  2 

To  translate  the  axes  to  the  maximum  point  set 

f  =  r  +  I,      s  -  s'  +  100.  j 

Substituting  in  (7), 

s'  +  100  =  -  16(/'  +  f)'  +  80(r  +  f), 

=  -  le^'*  -  m'  - 100  +  sot'  +  200, 

whence  s'  -  -  16f'*.  (10)1 

The  graph  of  this  equation  plotted  on  the  new  axes  is  identical  with  that 
of  (7)  plotted  on  the  old  axes.  The  figure  shows  the  graph,  which  is 
readily  plotted  on  the  new  axes.     It  passes  through  the  old  origin  sincej 


ALGEBRAIC  FUNCTIONS 


101 


Fig.  65. 


«  =  0  when  t  =  0.  From  the  symmetry  of  the  graph  wi^hresp^cito  tte'^ 
«'-axis  the  other  intercept  on  the  i-axis  h  t  =  5,  so  tliat  the  ball  returns  to 
the  hand  after  5  seconds. 
Hence  the  time  of  rising  is 
equal  to  the  time  of  falling. 
The  figure  also  shows  the 
graph  of  V  [equation  (9)]. 
As  y  is  a  linear  function  of 
t,  the  slope  -  32  gives  the 
acceleration,  which  is  con- 
stant. A  simultaneous  dis- 
cussion of  the  variation  of 
8  and  V  describes  the  motion 
fully.  At  the  start  s  =  0 
and  V  =  80.  As  t  increases, 
s  increases  but  the  velocity 
decreases,  until  when  t  -  5 
the  ball  reaches  its  maxi- 
mum height  of  100  feet  and 
the  velocity  becomes  zero.  The  ball  then  begins  to  fall,  since  the  velocity 
changes  sign,  and  its  height  decreases  to  zero  at  <  =  5,  when  the  velocity 
is  -  80,  i.e.,  80  feet  per  second  downward. 

The  position  of  the  parabola  which  is  the  graph  of  a  quadratic 
function  with  reference  to  the  x-axis  is  determined  by  the  dis- 
criminant (9(/)  page  xvii)  of  the  function,  for  the  discriminant 
determines  the  nature  of  the  roots  of  the  equation 

ax^  +  6x  4-  c  =  0, 

which  are  identical  with  the  zeros  of  the  function  and  which 
are  represented  by  the  intercepts  on  the  x-axis. 

If  the  discriminant  6^  —  4ac  is  positive  the  roots  are  real  and 
distinct,  if  the  discriminant  is  zero  the  roots  are  equal,  and  if 
the  discriminant  is  negative  the  roots  are  imaginary. 

It  follows  that  the  graph  of  a  quadratic  function  cuts  the 
a:-axis  twice  if  the  discriminant  is  positive.  If  the  discrimi- 
nant is  zero,  the  graph  has  but  one  point  in  common  with  the 
rc-axis,  and  hence  the  graph  is  tangent  to  the  x-axis.  The  point 
of  tangency  is  the  maximum  or  minimum  point.  If  the  dis- 
criminant is  negative,  the  curve  does  not  meet  the  a;-axis 
at  all. 


102 


ELEMENTARY  FUNCTIONS 


.  A  method  of  using  the  zeros  of  a  quadratic  function  in  con- 
structing the  graph  of  the  function  is  illustrated  in  the 


Fig.  66. 


Alternative  solution  for  the  example  above.     The  graph  of  equation  (7 
a  parabola  running  downward,  since  s  is  a  quadratic  function  of  t 

which  the  coefficient  of  f^  is 
negative. 

To  find  the  intercepts  on 
the  <-axis  set  s  =  0,  which 
gives- 16^2 ^80<  =  0.  Divid- 
ing by  -  16,  and  factoring, 
t{t  -  5)  =  0,  whence  t  =  0  or 
5.  Hence  the  origin  and  the 
point  A  (5,  0)  are  on  the 
graph.  These  intercepts  give 
the  time  of  starting  and  when 
the  ball  returns  to  the  hand. 

The  axis  of  symmetry  is 

vertical.       Hence     it     must 

bisect  OA,  and  it  is  therefore 

the  line  a:  =  f  =  2.5. 

Since  the  maximum  point  lies  on  the  axis  of  symmetry,  the  ball  reaches 

its  greatest  height  when  t  =  f  seconds.     Substituting  this  value  in  (7), 

the  maximum  height  is  -  100  +  200  =  100  feet. 

Hence  the  point  M(2.5,  100;  is  the  maximum  point. 
In  order  to  plot  the  curve  from  O  to  M  we  need  only  to  compute  the 
values  of  s  for  ^  =  1  and  2.    The  other  half  of  the  curve  is  obtained  by 
means  of  the  symmetry. 


EXERCISES 

1.  The  Theorem,  Section  34,  shows  that  the  graph  of  y  =  6a:  -  x^  is  a 
parabola,  with  vertical  axis,  running  downward.     Construct  the  graph: 

(a)  By  translating  the  axes,  as  in  Section  31. 

(b)  By  finding  the  maximum  point  as  in  Section  33  and  translating  the 
axes. 

(c)  By  finding  the  intercepts  on  the  a;-axis,  determining  the  axis  of 
symmetry  from  the  intercepts,  and  the  maximum  value  of  y  by  means  o: 
the  axis  of  symmetry. 

(d)  By  finding  the  values  of  y  to  be  excluded,  from  which  the  maximum 
value  of  y  may  be  found,  and  determining  the  axis  of  symmetry  by  means 
of  the  maximum  value. 

In  (c)  and  (d)  it  will  be  necessary  to  obtain  a  few  pairs  of  values  of  a>' 


ALGEBRAIC  FUNCTIONS  103 

and  y.     The  most  desirable  values  of  x  to  be  assumed  may  be  determined 
after  finding  the  coordinates  of  the  maximum  point. 

2.  Construct  the  graph  of  each  of  the  functions  below.  Find  the  axis 
of  symmetry,  and  the  maximum  or  minimum  point.  Find  the  zeros  of 
each  function,  and  state  how  they  are  represented  graphically. 

(a)  x^  +  2x-  8.  (b)  2x2  +  3x  -  2.  (c)  -  x^  +  3x. 

(d)  -  x2  -  X  +  2.  (e)  x2  -  4x  -i-  4.  (f)  -  x^  -  2x  -  1. 

(g)  x^  +  X  +  1.  (h)  4x2  +  2x  -  1.  (i)  _  6x2  +  X  +  5. 

(j)  x2  -  X  +  1.  (k)  5x2  +  2x  +  1.  (1)  -  3x2  4. 2x  -  2. 

3.  Without  solving  the  following  equations,  determine  whether  the  roots 
are  real  or  imaginary,  equal  or  unequal.  From  this,  what  can  be  said  of 
the  graph  of  the  quadratic  function  in  the  left-hand  member? 


(a)  x2  +  3x  -  5  =  0. 

(b)  2x2  -  3x  +  4  =  0 

(c)  4x2  +  12x  +  9  =  0. 

(d)  x2  -  |x  +  3  «  0. 

(e)  4x2  -  X  +  1  =  0. 

(f)   x«-7x-2  =  0. 

4.  A  special  quadratic  function  is  one  in  which  one  or  more  of  the  co- 
efficients are  zero.  Discuss  and  plot  the  graph  of  the  special  fimctions 
below,  and  state  the  special  properties  of  the  graph  which  distinguish  it 
from  the  general  case. 

(a)  x2  -  4.  (b)  x2  -  4x.  (c)  x2/3. 

(d)  9  -  x2.  (e)  3x  -  x2.  (f)    _  2x2. 

(g)  ax2  +  c.  (h)  ax2  +  6x.  (i)    axK 

5  If  /(x)  =  ax2  +  6x  +  c,  what  must  be  true  of  the  coefficients  if /(-  x)  = 
fix)? 

6.  If  a  ball  is  thrown  vertically  upward  with  a  velocity  of  50  feet  per 
second,  its  height  after  t  seconds  is  given  by  the  equation 

s  =  -  16^2  +  50^. 

Construct  the  graph.     Find  when  the  ball  will  be  highest,  how  high  it 
will  rise,  and  when  it  wiHYeturn  to  the  hand. 

7.  Construct  the  graphs  of  x  and  x2  from  x  =  0  to  x  =  1.  What  is  the 
numerical  difference  of  the  ordinates  of  points  on  these  graphs  with  the 
same  abscissa?  Construct  the  graph  of  this  difference,  and  find  the  value 
of  X  for  which  it  is  greatest.     What  is  the  greatest  difference? 

8.  A  Norman  window  consists  of  a  rectangle  (width  2x  and  height  y) 
surmounted  by  a  semi-circle.  If  the  perimeter  of  the  window  is  16  feet, 
find  the  area  in  terms  of  the  half  width  x,  and  construct  its  graph.  Find 
the  dimensions  which  make  the  area  (and  hence  the  amount  of  light  ad- 
mitted j  a  maximum. 


104  ELEMENTARY  FUNCTIONS 

9.  A  clifif  40  feet  high  overhangs  a  river.    A  man  on  the  cliff  throws  a 
Btone  vertically  upward  with  a  velocity  of  30  ft.  per  second.     If  the 
height  of  the  stone  above  the  cliff  after  t  seconds  is  given  by 
s  =  -  16f2  +  30t, 

find  how  high  the  stone  rises,  and  when  it  will  strike  the  water. 

10.  If  the  perimeter  of  a  rectangle  is  8  feet,  find  the  area  as  a  fmiction 
of  one  of  the  dimensions.  Construct  the  graph  of  the  area,  and  find  the 
dimensions  and  area  of  the  largest  rectangle  of  this  sort. 

11.  Solve  Exercise  10  if  the  perimeter  is  any  constant  p.  [The  parcel 
post  regulations  require  that  the  sum  of  the  length  and  girth  (greatest 
perimeter  of  a  section  at  right  angles  to  the  length)  of  a  package  shall  not 
exceed  6  feet.  By  means  of  this  Exercise,  what  can  be  said  of  the  shape 
of  the  largest  rectanglar  box  which  can  be  mailed?] 

12.  The  equation  of  the  path  of  a  ball  thrown  into  the  air  at  an  angle 
of  60°  with  the  horizon  with  a  velocity  of  32  feet  per  second  is 

y  =  i.7x  -  xym. 

Construct  the  graph,  find  the  greatest  height  attained,  and  where  the 
ball  will  hit  the  ground. 

13.  If  a  body  is  projected  vertically  upward  with  an  initial  velocity  of 
V  feet  per  second,  its  height  s  after  t  seconds  Is  given  by  the  equation 

s=  -  16^2  ^  vt. 

Find  how  long  the  body  will  rise  and  its  maximum  height.     Prove  that 
the  time  of  rising  equals  the  time  of  falling. 

14.  A  farmer  estimates  that  if  he  digs  his  potatoes  now  he  will  have  100 
bushels  worth  $1.25  a  bushel;  but  that  if  he  waits,  the  crop  will  increase 
16  bushels  a  week,  while  the  price  will  drop  Sff  a  week.  Find  the  value  of 
his  crop  as  a  function  of  the  time  in  weeks,  and  draw  the  graph.  When 
should  he  dig  to  get  the  greatest  cash  returns? 

15.  The  amount  of  wheat  obtained  per  acre  depends  on  the  intensity 
of  cultivation.  A  farmer  finds  that  he  can  cultivate  15  acres  with  suf- 
ficient intensity  so  that  the  return  will  be  30  bushels  per  acre;  20  acres 
BO  that  the  return  will  be  25  bushels  per  acre;  25  acres  so  that  the  return 
will  be  20  bushels  per  acre;  etc.  Find  the  law  giving  the  total  return  as 
a  function  of  the  number  of  acres  under  cultivation,  and  plot  the  graph. 
What  is  the  best  size  acreage  for  him  to  cultivate  in  order  to  get  the  lari 
est  gross  returns? 


35.  Empirical  Data  Problems.    The  table  of  values  of 
linear  function  is  such  that  if  the  successive  values  of  Ax 
equal  so  also  are  the  successive  values  of  Ay. 

Any  quadratic  function  has  a  somewhat  analogous  property, 
which  is  illustrated  in  the  table  for  y  =  2x^  +  Sx  +  4.     Tb 


ALGEBRAIC  FUNCTIONS 


105 


Ax 

X 

y 

t^y 

0 

4 

1 

5 

1 

9 

1 

9 

2 

18 

1 

13 

3 

31 

1 

17 

4 

48 

4^_ 

4 
4 
4 


values  of  Ax  being  equal,  those  of  Ai/,  sometimes  called  the 
jird  differences  of  2/,  are  not  equal.  But  the 
successive  differences  of  Ay,  called  the 
second  differences  of  y  and  denoted  by  A^y, 
are  equal.  Every  quadratic  function  pos- 
sesses this  property,  that  if  the  values  of  x 
are  such  that  the  successive  values  of  Ax  are 
equal,  then  the  su^ccessive  valves  of  the  second 
differences  oj  y,  A^y,  are  equxil. 

This  property  enables  us  to  tell  whether  a  given  table  of 
values  may  be  represented  approximately  by  a  quadratic 
function,  and  also  to  determine  the  coefficients  of  the  function. 

Example.  Find  a  quadratic  function  which  represents  approximately 
the  law  connecting  the  values  of  x  and  y  given  in  the  table. 

Inspection  of  the  values  of  x 
shows  that  the  successive  values 
of  Ax  are  equal.  Computing  the 
successive  values  of  Ay  and  A^y, 
it  is  found  that  the  latter  are 
nearly  equal.  Hence  the  values 
of  y  resemble  those  of  a  quadratic 
function. 
Now  let 

y  =  ax^  +  hx-\-c  (1) 

be   the   required   function.     The 
second  table  gives  the  values  of 
y,  Ay,  Ahf  computed  from  (1)  for 
the  values  of  x  in  the  given  table. 
If  the  given  values  of  y  were  exactly  the  values  of  some  quadratic  func- 
tion,  then    for  the  proper 
•values    of    a,   h,  c,   all  the 
numbers  of  one  table  would 
equal     the      corresponding 
numbers  in  the  other.     In 
particular,  the  average  val- 
ues of  y,  Ay,  and  Ahj  found 
in  the  second  table  would 
equal  those  in  the  first.    As 
it  is,  these  averages  are  equal 
for  values  of  a,  h,  c,  for  which 
(1)   represents  the  required 
law  approximately. 


X 

y 

Ay 

Ah/ 

2 

3.1 

8.1 

4 

11.2 

15.7 

7.6 

6 

26.9 

24.1 

8.4 

8 

51.0 

32.4 

8.3 

10 

83.4 

5  1  175.6 
35.12 

4  180.3 

3  1 24.3 

20.08 

8.1 

X 

y 

Ay 

AV 

2 

4a  +    26+    c 

12a  +  26 

4 

16a  +   45+   c 

20a  +  26 

8a 

6 

36a  +    6&+   c 

28a +  26 

8a 

8 

64o+   86+   c 

36a +  26 

Sa 

10 

100a +  106+   c 

5  1  220a  +  306  +  5c 

4  1  96a  +  86 

31  24a 

44a  +    66+    c 

24a  +  26 

8a 

106 


ELEMENTARY  FUNCTIONS 


Equating  the  average  values  of  Ahj,  Sa  =  8.1,  whence  a  =  1.01. 
Equating  the  average  values  of  Ay,  24a  +  26  =  20.08.     Substituting  the 
value  of  a,  and  solving  for  6,  we  get  6  =  -  2.08. 

Equating  the  average  values  of  y,  44a  -f  66  +  c  =  35.12.  Substituting 
the  values  of  a  and  6,  and  solving  for  c,  we  get  c  =  3.16.     Substituting  the 

values  of  a,  6,  c,  in  (1), 
_x  Observed  y  Computed  y\  Error  %  error  the  required  approxima- 
2  3.1  3.04  -0.06      -1.9       tion  of  the  law  is 

4         11.2  11.00         -0.20      -1.8       2/ =  1.01x2 -2.08X+ 3.16 

6         26.9  27.04         +0.14     +0.5  The     accuracy     with 

8         51.0  51.16         +0.16+0.3       which    this    relation    ex- 

10         83.4  83.36         -0.04      -0.05     presses  the  law  is  indi- 

cated in  the  table. 

If  it  is  desired  to  find  the  equation  of  a  parabola  with  verti- 
cal axis  which  passes  through,  or  near,  several  points  whose 
coordinates  are  given,  the  method  used  in  the  example  may  be 
employed  even  though  the  values  of  Ax,  and  hence  also  those 
of  A^y,  are  not  equal.     At  least  three  points  must  be  given. 

EXERCISES 

1.  Find  the  equation  of  the  parabola  with  vertical  axis  which  passes 
through  the  points  (1,  0)  ,(3,  10),  (5,  28).  Construct  the  graph  and  check 
the  result. 

2.  In  sinking  a  deep  mine,  new  material  as  well  as  labor  must  be  in- 
vested each  day,  and  the  required  depth  cannot  be  determined  in  advance 
with  accuracy.  A  company  which  has  set  aside  $100,000  for  the  cost  of 
sinking  a  shaft  finds  the  total  capital  invested  as  the  work  increases  as 

rr..       .  .,  1  n     1       o  o        given  in  the  table.    Find 

Time  m  months  i  n     i       o  o 


Investment  in  thousands  |  4,  12,  26.1,  46.1, 


0,    1,     2, 


the  law  giving  the  invest- 


ment as  a  function  of  the 
time.  If  the  work  continues  to  progress  according  to  the  same  law, 
when  must  the  work  be  completed  if  the  cost  is  not  to  exceed  the  amount 
set  aside? 

3.  A  contractor  agreed  to  build  a  breakwater  for  $125,000.  After  spend- 
ing $33,000  as  indicated  in  the  table,  he  threw  up  the  job  without  receiv- 
ing any  pay,  because  he  estimated  that  the  cost  would  increase  according  to 
the  same  law,  and  that  it  would  require  six  months  more  to  complete  the 
work.     How  much  would  he  have  lost  if  he  had  finished  the  breakwater? 


Time  in  months 


0, 


1, 


2, 


Investment  in  thousands  I  5,  6.1,  11.2,  20.2,  33.1 

4.   An  electric  conductor  gives  out  a  definite  amount  of  current  in  every 
mile  of  its  length.     Let  x  be  the  distance  of  any  point  in  miles  from  the 


ALGEBRAIC  FUNCTIONS  107 

end  of  the  line  remote  from  the  generator,  and  y  the  voltage  there.  As- 
Buming  that  the  law  is  of  the  form  y  =  ax^  +  5,  determine  the  law  from  the 
table.     What  is  the  voltage  at  a  point  2.5  miles  from  the  end? 

Hint:  The  value  of  a  may  be  de- 
:  I  0,  1,  2,  3,  4  termined  by  the  general  method; 
^1  200,  212.5,  250,  312.5,  400    after  findmg  a,  substitute  its  value  in 

y  =  ax^  -\-h.  Then  a  value  of  6  may 
be  determined  by  substituting  any  pair  of  values  of  x  and  y.  A  better 
value  of  h  is  obtained  by  determining  its  value  for  each  pair  of  values  of 
X  and  y  and  averaging  the  results  (compare  the  method  of  finding  h  in 
Section  27  after  m  is  determined). 

36.  The  Function  x^»  Among  the  functions  represented  by 
a;"  which  may  be  obtained  by  assigning  a  numerical  value  to  n 
may  be  mentioned  the  following: 

x^,  for  n  =  2. 

x^j  for  n  =  3. 

xi  =  =fc  ^s/x,  for  n  =  J. 

x^  =  -s/x,  for  n  =  \. 

x~^  =  1/x,  f or  n  =  —  1. 

If  no  numerical  value  is  assigned  to  n,  the  sjnnbol  x^  may 
represent  either  the  totality  of  all  functions  obtained  by  assign- 
ing a  numerical  value  to  n,  or  a  particular,  but  unassigned,  one 
of  these  functions. 

The  function  x",  sometimes  called  the  power  function,  is  con- 
sidered in  the  following  sections. 

37.  Tables  of  Squares,  Cubes,  Square  Roots,  Cube  Roots, 
and  Reciprocals.  These  tables  are  extensive  tables  of  values 
of  the  function  a;"  for  n  =  2,  3,  |,  J,  and  -  1.  They  are  labor- 
saving  devices,  for  from  them  we  can  find,  for  example,  the 
square  of  a  number  without  the  labor  of  multiplying  it  by  it- 
self. We  shall  use  Huntington's  Four  Place  Tables,  Unabridged 
Edition. 

Tables  of  sqicares  and  cubes.  The  square  of  a  number  n, 
between  1  and  10,  may  be  found  from  the  table  on  page  2. 
In  order  to  economize  room,  the  table  is  not  arranged  in  two 
long  columns  of  rows.  Instead,  the  first  digits  in  n  are  given 
in  the  border  on  the  left  and  the  last  digit  in  the  border  at  the 


108  ELEMENTARY  FUNCTIONS 

top  of  the  table.  To  find  the  square  of  1.56,  look  in  the  row  in 
which  1.5  stands  on  the  left,  under  "  n,"  and  in  the  column 
headed  "  6."  Here  we  find  2.434,  which  is  the  square  of  1.56 
to  four  figures.  The  exact  square,  obtained  by  multiplication, 
is  2.4336,  which  is  nearer  2.434,  as  given  in  the  table,  than  to 
2.433.  The  lack  of  exactness  in  the  tables  is  usually  im- 
material, for  in  most  of  the  applications  of  mathematics  three  or 
four  figure  accuracy  is  all  that  is  desired^  and  in  many  it  is  all 
that  can  be  attained.  Thus  if  1.56  is  the  side  of  a  square,  ob- 
tained by  measurement,  the  area  is  1.56^  =  2.43.  Only  three 
figures  are  retained  in  the  area  since  the  product  cannot  con- 
tain more  significant  figures  than  the  factors  (Section  26).  Foi^ 
such  an  example  the  table  of  squares  is  more  accurate  than' 
necessary. 

To  find  the  squares  of  numbers  greater  than  10,  or  less  than 
1,  we  use,  respectively,  the  relations 

(10n)2,=  1007i2        and        {n/lOy  =  n^lOO. 

Since  multiplication  by  10  or  100  shifts  the  decimal  point 
one  or  two  places  to  the  right,  respectively,  while  division 
shifts  it  to  the  left,  we  have  the  rule  given  at  the  top  of  the 
table:  "  Moving  the  decimal  point  one  place  in  n  is  equivalent 
to  moving  it  two  places  in  n^." 

Repeated  application  of  this  rule  enables  us  to  find  the 
square  of  any  number.  Thus  to  find  247^,  we  find  from  the 
table  2.472,  =  6.10,  whence,  applying  the  rule  twice,  247^  = 
61,000.     Sinularly,  0.247^  =  0.0610. 

The  table  of  cubes,  on  page  4,  is  very  similar  to  the  table 
of  squares.  The  rule  at  the  top  of  the  table  for  shifting  the 
decimal  follows  from  the  relations: 

{lOnY  =  lOOOn'        and        (n/ioy  =  nVlOOO. 

Tables  of  square  roots  and  cube  roots.  The  table  of  square 
roots  on  page  3  is  separated  into  two  parts  which  give  the 
square  roots  of  numbers  from  0.1  to  1  and  from  1  to  10.  The 
reason  for  this  lies  in  the, rule  for  shifting  the  decimal  point. 


ALGEBRAIC   FUNCTIONS  109 


Since  VlOOn  =  lOVn       and        X/Tno 


10" 


we  have  the  rule  at  the  top  of  the  table:  *'  Moving  the  decimal 
point  two  places  in  n  is  equivalent  to  moving  it  one  place  in  V^" 

If  n  is  greater  than  10  or  less  than  0.1,  by  moving  the  point 
two  places  at  a  time,  to  the  left  or  right,  respectively,  we  will 
ultimately  obtain  a  number  in  which  the  decimal  point  either 
precedes  or  follows  the  first  significant  figure.  In  the  former 
case  the  square  root  is  found  in  the  upper  part  of  the  table, 
and  in  the  latter  case  in  the  lower  part.  The  square  root  of 
the  original  number  is  then  found  by  re-shifting  the  decimal 
point  in  accordance  with  the  rule. 

For  example,  to  find  the  square  root  of  35,700,  we  shift  the 
point  two  places  to  the  left  twice  in  succession,  obtaining  3.57. 
From  the  lower  part  of  the  table  we  obtain  V3.57  =  1.889. 
Then  applying  the  rule  twice,  shifting  the  point  to  the  right, 
\/35700  =  188.9. 

To  find  the  square  root  of  0.0024,  we  shift  the  point  two 
places  to  the  right,  getting  0.24.  From  the  upper  part  of  the 
table,  VO^  =  0.490,  whence,  by  the  rule,  VO.0024  =  0.0490. 
Notice  that  the  zero  following  the  nine  is  retained  in  order  to 
show  the  value  of  the  radical  to  three  figures. 

The  table  of  cube  roots  on  page  5  is  divided  into  three  parts 
because  in  moving  the  decimal  point  in  any  number  three 
places  at  a  time  (see  rule  at  the  top  of  the  table),  we  ultimately 
obtain  a  number  in  which  the  point  either  follows  the  first 
significant  figure,  precedes  it  immediately,  or  is  followed  by  a 
single  cipher. 

Table  of  reciprocals.  The  reciprocal  of  n,  1/n,  differs  from 
n^,  n^,  Vn,  \/n,  in  that  as  n  increases  the  reciprocal  decreases. 
For  a  reason  which  will  appear  in  Section  42,  it  is  desirable  to 
have  all  the  tables  so  arranged  that  the  numbers  in  the  body  of 
the  table  increase  as  we  read  down  the  page,  and  from  left  to 
right.  The  table  of  reciprocals  on  page  7  is  so  arranged.  The 
values  of  n,  given  in  the  border  of  the  table,  increase  as  we 
read  up  and  to  the  left. 


110  ELEMENTARY  FUNCTIONS 

The  rule  for  shifting  the  decimal  point  at  the  top  of  the 
table  follows  from  the  fact  that  the  product  of  n  and  its  re- 
ciprocal is  imity.  Hence  if  one  is  multipHed  by  10,  the  other 
must  be  divided  by  10. 

To  find  1/436,  for  example,  we  read  up  the  table,  on  the  right, 
until  we  come  to  4.3,  then  over  to  the  left  until  we  are  in  the  col- 
umn headed  6,  where  we  find  0.2294,  which  is  the  value  of  1/4.36. 
Then  by  the  rule  for  shifting  the  point,  1/436  =  0.002294. 

In  using  any  one  of  these  tables  the  position  of  the  decimal 
point  may  frequently  be  determined  by  inspection,  and  it  is 
well  to  check  the  result  obtained  from  the  table. 

EXERCISES 


1.  Find  3.242,  ^0.47,  V4.72,  7.43,  ^o.0235,  ^0.24,  ^1.84,  1/24. 

2.  Find  the  squares  of  25.4,  0.86,  3540,  0.0043. 

3.  Find  the  square  roots  of  59,  590,  4300,  0.000382. 

4.  Find  the  cubes  of  54,  0.317,  53200,  0.0000371. 

6  Find  the  cube  roots  of  1540,  470,  18.3,  0.0048,  0.0000259. 

6.  Find  the  reciprocals  of  23.4,  0.478,  532,  0.0074.     

7.  Find  the  value  of  0.07S  0.06^,  V'14.23,  1/234^,  ^0.02472. 

8.  Find  the  hypothenuse  of  a  right  triangle  whose  sides  are  61  and  74. 

9.  Find  the  mean  proportional  between  6  and  34. 

38.  Graph  of  jc",  n>l.  It  will  be  seen  in  this  section,  and 
those  following,  that  a  part  of  the  graph  of  a:"  hes  in  the  first 
quadrant,  and  that  the  remaining  part  may  be  found  by  means 
of  the  symmetry  of  the  curve.  Hence  particular  attention 
should  be  made  to  fix  in  mind  the  various  forms  of  the  part  of 
the  graph  in  the  first  quadrant.  The  general  appearance  of 
this  part  of  the  graph  varies  according  as  n>l,  0<n<l, 
or  n<0,  so  that  these  cases  are  considered  separately.  These 
groups  of  values  include  all  values  of  n  except  n  =  0  and  n  =  1, 
which  separate  the  groups,  and  which  are  exceptional  in  that 
they  are  the  only  values  for  which  the  graph  is  a  straight  line. 

n  =  0.  If  n  =  0, 2/  =  x""  becomes  2/  =  1,  since  x^  =  1,  whose  graph 
is  the  strai^t  line  parallel  to  the  x-axis  and  one  unit  above  it. 

w  =  1.  For  w  =  1  we  have  y  =  x,  whose  graph  is  the  straight 
line  through  the  origin  whose  slope  is  unity,  that  is,  the  bisector 
of  the  first  and  third  quadrants. 


ALGEBRAIC   FUNCTIONS 


111 


n>l.     In  this  case  the  graph  of 

y  =  x^  (1) 

is  tangent  to  the  x-axis  at  the  origin  (0,  0),  rises  to  the  right 
and  passes  through  the  point  (1,  1),  at  which  the  slope  of  the 
tangent  Hne  is  n.  That  it  passes  through  these  points  is  seen 
by  substituting  their  coordinates  in  (1). 

To  find  the  slope  of  the  fine  tangent  to  the  graph  at  any  point, 
rhen  w  is  a  positive  integer,  replace  a;  by  x  +  Aa:  and  yhyy  +  Ay 
(1),  which  gives  y  +  Ay  =  (x  +  Ax)". 

Expanding  the  right  hand  member  by  the  binomial  theorem, 
n(n  —  1) 


+  Ai/  =  X"  +  nx"-^Ax  + 
Subtracting  (1)  from  (2), 

n(n  —  1) 


^n-2^^2   _|. 


+  Ax"  (2) 


X«-2Ax2  + 


Ay  =  nx"~^Ax  + 
Dividing  by  Ax, 


•  +Ax". 


+  Ax»-^ 


Passing  to  the  limit  as  Ax  approaches  zero,  the  slope  of  the 
]ent  line  at  any  point  is 

m  =  nx"-i.  (3) 

This  proof  assumes  that  n  is  a  posi- 
tive integer,  but  in  Chapter  VI  it  will 
be  shown  that  the  result  holds  for 
fractional  and  negative  values  of  n. 

At  the  origin  x  =  0,  so  that,  by 
(3),  m  =  0  if  ri  >1.    Hence  the  tan- 
gent at  the  origin  is  the  x-axis. 
At  the  point  (1,  1)  the  slope  of  the 
,'  tangent  line  is  m  =  n.   Hence  the  larger 
t  the  value  of  n  the  steeper  the  curve  is 
at  this  point,  from  which  it  follows 
that  it  must  be  flatter  near  the  origin. 

If  n  is  a  positive  integer  greater  than  unity,  the  part  of  the 
graph  of  X"  in  the  first  quadrant  is  then  very  much  like  the 
graph  of  x2. 


112 


ELEMENTARY  FUNCTIONS 


If  n  is  a  positive  fraction  greater  than  unity  the  part  of  the 
graph  in  the  first  quadrant  has  the  same  general  appearance. 
To  see  this,  let  r,  s,  and  t  be  three  values  of  n  such  that  r<s<t. 
For  a  positive  value  of  x  the  value  of  x*  will  lie  between  a^*"  and 
x^,  and  hence  the  part  of  the  graph  of  x*  in  the  first  quadrant 
lies  between  the  graphs  of  x''  and  x*  (this  holds  for  all  values  of 
r,  s,  and  t).  For  example,  the  graph  of  x^  lies  between  the 
graphs  of  x  and  x^;  that  of  x^  between  those  of  x^  and  a^;  etc. 

The  figures  show  the  graphs  of  x^,  x^,  and  x^,  which  are  sym- 
metrical with  respect  to  the  ?/-axis,  the  origin,  and  the  a;-axis 


respectively.  The  symmetry  of  the  la^t  curve  is  seen  by 
writing  y  =  xl  in  the  form  y'^  =  a^,  and  applying  Theorem 
SB,  page  24.  The  graph  of  x^,  w>l,  always  resembles  one  of 
these  curves.  Thus  the  graph  of  x*^  is  very  much  like  a  parabola, 
but  it  is  flatter  near  the  origin  and  steeper  elsewhere. 

The  a;-axis  is  tangent  to  the  graph  of  a^  at  the  origin.  This 
tangent  differs  from  any  we  have  encountered  hitherto  in  that 
it  crosses  the  curve  at  the  point  of  tangency. 

The  graph  of  x^  is  remarkable  in  that  it  has  a  sharp  point 
a+  the  origin.  It  differs  from  other  curves  we  have  studied 
in  detail  in  that  vertical  lines  to  the  right  of  the  2/-axis  cut  it 


ALGEBRAIC  FUNCTIONS 


113 


9  {y,^) 


P{x*y) 


are 


twice,   corresponding  to  the   fact   that  x'  =  ±Va^   has  two 
values  for  each  positive  value  of  x. 
39.    Graph  of  x",  0<n<l.    Graphs  of  Inverse  Functions. 

Let  us  first  consider  n  =  \,  or  the  function  x^.    If  we  set  y  =  xh 

and   solve   for  x,  we   get  x  =  y^. 

This    differs    from    the    equation 

y  =  x^  only  in  that  x  and  y  have 

been  interchanged.    Hence  if  (x,  y) 

is  a  point  on  the  graph  of  either 

equation,  the  point  {y,  x)  is  on  the 

graph   of    the   other.    But    these 

points  are  symmetrical  with  respect 

to  the  bisector  of  the  first  and  third 

quadrants,  as  may  be  established 

from  the  figure.    Hence  the  graphs 

of  the  two  equations,  that  is,  the  graphs  of  x^  and  a;^, 

symmetrical  to  each  other  with  respect  to  this  bisector. 

The  graph  of  x^  may  therefore  be  constructed  as  follows: 
Construct  the  graph  of  rc^  and  the  bisector  of  the  first  and 
third  quadrants.  Choose  a  number  of 
points  on  the  graph  of  x^,  and  con- 
struct the  points  symmetrical  to  them 
with  respect  to  this  bisector.  Draw 
a  smooth  curve  through  the  points  so 
obtained. 

We  may  now  get  properties  of  the 
function  x*  by  interpreting  its  graph. 
For  example,  since  the  graph  of  x^  is 
symmetrical  with  respect  to  the  y-sods 
that  of  x^  is  symmetrical  with  respect 
to  the  a;-axis,  and  hence  to  each  value  of  x  there  correspond 
I  two  values  of  x^  which  are  equal  numerically  but  differ  in  sign. 
;  And  since  no  part  of  the  graph  lies  to  the  left  of  the  2/-axis 
!  (why?),  the  function  is  imaginary  if  x  is  negative.  What 
t  other  properties  may  be  obtained  in  this  way? 

If  two  curves  are  symmetrical  to  each  other  with  respect  to 
i    a  Une  they  are  congruent,  for  one  may  be  brought  into  co- 


Fig.  60. 


m 


114 


ELEMENTARY  FUNCTIONS 


incidence  with  the  other  by  rotating  the  plane  about  the  Une 
through  180°.     Hence  the  graph  of  x^  is  a  parabola. 

Since  we  obtained  the  equation  y  =  x^  hy  solving  y  =  x^  for 
X  and  then  interchanging  x  and  y,  the  functions  x^  and  x*  are 
inverse  functions  (page  40) .  The  graphical  considerations  above 
may  be  applied  to  any  two  inverse  functions,  and  hence  we 
have  the 

Theorem.  The  graphs  of  inverse  functions  are  symmetrical 
to  each  other  with  respect  to  the  bisector  of  the  first  and  third  quad- 
rants. 

If  the  graph  of  any  function  is  given,  then  the  graph  of  the 
inverse  function  may  be  obtained  readily  by  this  theorem. 
The  distinction  between  two  curves  symmetrical  to  each  other 
with  respect  to  a  line  and  a  single  curve  which  is  symmetrical 
with  respect  to  a  line  should  be  noted. 

To  find  the  inverse  of  x",  set  y  =  x"",  and  interchange  x 
and  y  which  gives  x  =  ?/";  solving  for  y  we  get  y  =  3^.  Hence 
the  inverse  of  x"  is  x^,  and  the  graphs  of  these  functions  are 


Fig.  61. 


symmetrical  to  each  other  with  respect  to  the  bisector  of  the 
first  and  third  quadrants. 

By  means  of  this  symmetry,  for  example,  the  graph  of  x* 


ALGEBRAIC  FUNCTIONS  115 

y  be  obtained  from  that  of  x^j  and  the  graph  of  x^  from 

at  of  x^. 

The  graph  of  re",  if  0<n<l,  always  resembles  the  graph  of 
one  of  the  functions  x^,  x*,  and  x^.  In  the  first  quadrant,  the 
graph  is  tangent  to  the  2/-axis  at  the  origin,  rises  to  the  right, 
and  passes  through  the  point  (1,  1),  at  which  the  slope  of  the 
tangent  Une  is  n.  Hence  the  smaller  the  value  of  n,  the  less 
rapidly  the  graph  rises  at  this  point,  from  which  it  follows  that 
it  must  be  steeper  near  the  origin.  The  remaining  part  of  the 
graph  may  be  determined  by  means  of  the  symmetry  of  the 
curve  with  respect  to  one  of  the  axes  or  the  origin. 

EXERCISES 

1.  Plot  the  graph  of  a:"  for  each  of  the  values  of  n  given  below,  using 
the  same  axes. 

(a)  n  =  2,  4,  6.  What  can  be  said  of  the  graph  if  n  is  an  even  positive 
integer? 

(b)  n  =  3,  5.  What  can  be  said  of  the  graph  when  n  is  an  odd  positive 
integer? 

(c)  n  =  1,  2,  3,  \,  \,  f,  f,  using  as  large  a  scale  as  possible. 

2.  Find  the  slope  of  the  tangent  line  at  any  point  to  the  graph  of  a;** 
for  n  =  2,  3,  4,  5.  Find  and  tabulate  the  slope  for  each  graph  at  the 
points  for  which  x  =  0,  0.3,  0.5,  0.8,  1,  2.  As  n  increases,  what  can  be 
said  of  the  slope  of  the  tangent  line  at  a  point  near  the  origin?  Remote 
from  the  origin? 

3.  On  separate  axes  sketch  the  graphs  of  x"  for  n  =  f ,  f ,  f .  Is  there 
much  difference  between  the  parts  of  these  curves  in  the  first  quadrants? 

4.  Find  the  inverse  of  each  of  the  functions  in  Exercise  3,  and  sketch 
their  graphs. 

5.  Determine  the  symmetry  of  x^  when  n  is  a  fraction  p/q,  if  (a)  p  is 
even  and  q  is  odd;  (b)  p  is  odd  and  q  is  even;  (c)  p  and  q  are  both  odd. 
What  values  of  n  give  typical  forms  of  the  graph  in  these  three  cases  if 
p/g>l?    Ifp/?<1? 

6.  Plot  the  graph  of  x^,  find  the  inverse  function,  and  plot  its  graph. 

7.  Would  it  be  easy  to  build  a  table  of  values  for  a:*?  Construct  its 
graph  by  first  finding  the  inverse  function. 

8.  Plot  the  graphs  of  the  following  functions,  in  each  case  finding  the 
inverse  function  and  its  graph. 

(a)  2/  =  a;2  -  2.  (b)  2/  =  x^  +  2a;.  (c)  y  =  -  x^  +  4x  -  4. 

9.  Using  Huntington's  Tables,  construct  the  graphs,  on  the  same 
axes,  of  X,  x^,  x^,  x%  x^,  taking  x  =  0.2,  0.4,  0.6,  .  .  .  ,  1.4.  Use  as  large  a 
scale  as  possible  and  plot  the  parts  of  the  graphs  in  the  first  quadrant  only. 


116  ELEMENTARY  FUNCTIONS 

10.  Using  Huntington's  Tables,  construct  a  table  of  values  of  x^  for  x  = 
0.05,  0.10,  0.15,  0.20,  0.25,  obtaining  the  values  of  the  function  to  two 
decimal  places.  Construct  the  graph  from  a;  =  0  to  a;  =  0.25  on  as  large  a 
scale  as  possible. 

11.  Plot  the  graph  of  x^  -  Ax.  On  the  same  axes  sketch  the  graph  of 
the  inverse  function,  and  state  several  of  its  properties  by  interpreting 
its  graph.     Can  you  find  the  inverse  function? 

Note.  In  finding  the  inverse  of  a  function,  it  is  necessary  to  solve  an 
equation.  At  any  stage  of  mathematical  development,  the  solution  of  an 
equation  may  be  impossible  by  means  of  functions  already  studied.  Such 
an  equation  defines  a  new  function,  whose  fundamental  properties  are  de- 
termined by  the  equation. 

Thus  it  is  impossible  for  a  student  beginning  algebra  to  solve  the  equa- 
tion y^  =  x  for  y.  It  is  first  necessary  that  he  should  become  acquainted 
with  the,  to  him,  new  function  x^  =  =t\/x. 

A  person  unacquainted  with  the  solution  of  cubic  equations  cannot  find 
the  inverse  of  the  cubic  function  in  Exercise  11.  But  the  theorem  on 
the  graphs  of  inverse  functions  enables  us  to  get  the  graph  and  some 
of  its  properties,  even  though  we  do  not  know  the  function.  This 
point  of  view  will  be  useful  later  in  studying  certain  transcendental 
functions. 

The  inverse  of  a  cubic  or  biquadratic  function  (page  39)  is  an  algebraic 
function,  but  the  inverse  of  a  polynomial  of  higher  than  the  fourth  degree 
is  usually  transcendental. 

12.  What  is  the  inverse  of  the  function  l/rc?  What  therefore  can  be 
said  of  the  symmetry  of  the  graph  of  the  function? 

13.  Show  that  the  graph  of  an  equation  is  symmetrical  with  respect  to 
the  line  y  =  xii  the  equation  is  unchanged  when  x  and  y  are  interchanged. 
Plot  the  graph  of  xy  -2x  -ly  =  0. 

14.  What  is  the  form  of  an  equation  in  x  and  y  if  it  defines  a  function 
of  X  which  is  its  own  inverse? 

15.  Plot  the  graphs  of  x^  and  x^  on  the  same  axes.  Show  how  the  fol- 
lowing facts  are  illustrated,     (a)  Wiy  =  x.     (b)  V^  =  x. 

16.  Plot  the  graphs  of  x^  and  xl  on  the  same  axes^  Show  how  the  fol- 
lowing facts  are  illustrated,     (a)  (\^xy  =  x.     (b)  \/x^  =  x. 

17.  For  what  value  of  x  does  x^  increase  at  the  same  rate  as  a:?  For 
what  value  does  x^  increase  less  rapidly?  more  rapidly?  interpret  the  re- 
sults graphically. 

18.  Find  the  value  of  x  for  which  the  tangent  lines  to  the  graphs  of  x* 
and  x^  are  parallel.  For  what  values  of  x  is  the  graph  of  x^  "flatter "  than 
that  of  x2?  for  what  values  is  it  steeper? 

19.  The  horse  power  of  a  gas  engine  is  sometimes  determined  by  the 
equation  H.P.  -  ihi/2.5,.  where  d  is  the  diameter  of  a  cylinder  and  n  is 
the  number  of  cylinders. 


I 


ALGEBRAIC  FUNCTIONS 


117 


"~" 

"t" 

X 

^ 

/ 

* 

/ 

J 

-: 

-■ 

- 

0 

s 

r-J 

_ 

~— 

^ 

f  f  ^ 

i 

/ 

/ 

' 

1 

_ 

Fig.  62. 


(a)  Plot  the  graph  for  four-cyhnder  engines,  taking  d  =  2.5,  3, 3.5,  4.   On 
the  same  axes,  plot  the  graphs  for  six-,  eight-  and  twelve-cylinder  engines. 

(b)  Plot  the  graph  for  d  =  3,  taking  n  =  4,  6,  8,  12.    On  the  same  axes, 
plot  the  graph  if  d  =  2.5,  or  d  =  4. 

40.   Graph  of  x",  n<0.    Graphs  of  Reciprocal  Functions. 

Consider  first  n  =  -  1,  the  function  x~^  =  1/x.  Let  y  =  l/x. 
Replacing  a;  by  -  x  and  yhy  -  y,  and  changing  the  signs  of 
both  sides  of  the  equation,  we  obtain  the  given  equation. 
Hence  the  graph  is  symmetrical  with 
respect  to  the  origin,  and  the  table  of 
values  need  include  only  positive  values 
of  X, 

Since  the  function  becomes  infinite 
as  X  approaches  zero,  the  2/-axis,  x  =  0, 
is  an  asymptote.  Solving  y  =  l/x  for 
X  we  get  X  =  \/y,  from  which  it  fol- 
lows that  the  x-axis,  2/  =  0,  is  also  an 
asymptote. 

The  figure  shows  the  graph, 
which  appears  to  be  symmet- 
rical with  respect  to  the  fine 
y  =  X,  the  bisector  of  the  first  and  third  quadrants.  That  this 
is  the  case  is  seen  as  follows:  The  equation  y  =  l/x,  or  xy  =  I, 
is  unchanged  if  x  and  y  are  interchanged.  Hence  if  (x,  y)  is 
on  the  graph,  so  also  is  (y,  x).  But  these  points  are  symmet- 
rical with  respect  to  the  fine  y  =  x  (Section  39),  and  therefore 
^ihe  graph  is  also. 

The  graphs  of  x  and  l/x  illustrate  the  following  properties 
of  any  two  reciprocal  variables : 

j     I.   Reciprocal  variables  have  the  same  sign.    For  both  graphs 
Eire  above  the  x-axis,  or  both  are  below,  for  any  value  of  x. 

II.  If  a  variable  increases,  the  reciprocal  decreases,  and  vice 
')ersa.  For  as  x  increases,  that  is,  as  the  graph  of  x  rises,  the 
?raph  of  l/x  falls;   and  as  x  decreases,  the  graph  of  l/x  rises. 

III.  If  the  numerical  value  of  a  variable  approaches  and  be- 
comes unity,  so  also  does  that  of  the  reciprocal.  For  the  graphs 
ntersect  at  the  points  (1,  1)  and  (-  1,  -  1). 


X 

0,  i  i  i,  1,  2,  3,  4, 

l/x 

»,  4,  2,  1,  1,  h  I  i 

118 


ELEMENTARY  FUNCTIONS 


IV.   //  a  variable  approaches  zero,  its  reciprocal  becomes  in- 
finite,  and  vice  versa.    For  both  axes  are  asymptotes. 


n  =  -  2,    the  function   x~^  =  1/x^.    The   function    1/x^ 


IS 


called  the  reciprocal  of  the  function  x^  in  accordance  with  the 
Definition.    Two  functions  are  said  to  be  reciprocal  if  their 
product  is  unity. 

The  following  properties  of  the  graphs  of  reciprocal  func- 
tions are  proved  by  the  hke  numbered  facts  above: 

la.  Corresponding  parts  of  the  graphs  of  reciprocal  functions 
lie  on  the  same  side  of  the  x-axis. 

Ila.  //  the  graph  of  a  function  rises,  the  graph  of  the  reciprocal 
function  falls,  and  vice  versa. 

Ilia.  If  the  graph  of  a  function  approaches  a  point  on  either 
of  the  lines  y  =  d=  1,  the  graph  of  the  reciprocal  function  ap- 
proaches the  same  point  from  the  opposite  side  of  the  line.  The 
points  of  intersection  of  the  graphs  lie  on  these  lines. 

IVa.  If  the  graph  of  a  function  crosses,  or  is  tangent  to,  the 
X-axis  at  the  point  (a,  0),  the  line  x  =  a  is  an  asymptote  of  the 
graph  of  the  reciprocal  function,  and  vice  versa. 

If  the  graph  of  a  function  has  been  plotted,  these  considera- 
tions enable  us  to  sketch,  roughly,  the  graph  of  the  reciprocal 
function,  without  building  a  table  of  values. 

The  general  form  of  the  graph  o: 
1/x^  may  be  obtained  from  that  of 
as  follows: 

At  the  extreme  left  of  the  figure,  thi 
graph  of  x^  Hes  above  the  rr-axis,  and  i 
falHng.  Then  the  graph  of  l/x"^  \i 
above  the  x-axis,  and  is  rising  (by  la 
and  Ila).  As  x  increases  up  to  a;  =  —  1, 
the  graph  of  x"^  falls  until  it  reaches 
the  line  y  =  I,  and  hence  the  graph  of 
l/x^  rises  until  it  reaches  this  line 
(by  Ilia).  From  these  facts  we  can  sketch  the  graph  of  1/a^ 
from  the  extreme  left  of  the  figure  up  to  a;  =  -  1. 

As  x  increases  from  -  1  to  0,  the  graph  of  x^  falls  and  be- 


V 

8 

4  zwi 

----vM---- 

•'1-^r  Mir 

Fig.  63. 


ALGEBRAIC   FUNCTIONS 


119 


comes  tangent  to  the  x-axis.  Hence  the  graph  of  Xjx^  rises 
indefinitely,  and  approaches  the  2/-axis  as  an  asymptote  (by 
Ila  and  IVa). 

As  X  increases  from  0  to  1,  the  graph  of  x^  rises  to  the  Une 
y  =  1,  and  hence  that  of  1/x^  falls  to  this  line.  As  x  increases 
from  1  on,  the  graph  of  x^  rises  indefinitely,  and  that  of  1/x^ 
continues  to  fall,  but  remains  above  the  x-axis.  The  right 
hand  half  of  the  figure  might  also  be  obtained  from  the  sym- 
metry with  respect  to  the  y-axis. 

Notice  that  the  graph  of  l/x^  hes  between  the  graph  of  1/x 
and  the  x-axis  to  the  right  of  x  =  1,  while  between  x  =  0  and 
x  =  I,  the  graph  of  1/x  lies  between  it  and  the  2/-axis. 

41.  Summary  of  Graph  of  x\    The  figure  shows  the  typical 
form  of  the  graph  in  the  first  quadrant  for  each  of  the  cases 
n>l,  0<n<l,  and  n<0.    This  part  of  the  graph  always  lies 
in   the   parts   of   the 
plane    shaded     alike.     V'' 
The  following  proper- 
ties are  noteworthy: 

The  graph  passes 
through  the  point  (1, 
1),  at  which  the  slope 
of  the  tangent  fine 
is  n. 

The  rest  of  the 
graph  may  be  deter- 
mined from  the  part 
in  the  first  quadrant 
by  means  of  the  sym- 
metry with  respect  to 
one  of  the  axes  or  the 
origin. 

The  following  remarks  apply  only  to  the  part  of  the  graph 
in  the  first  quadrant. 

n>l.  The  graph  is  tangent  to  the  x-axis  at  the  origin,  and 
rises  as  it  runs  to  the  right.  The  larger  the  value  of  n,  the 
flatter  the  graph  is  near  the  origin,  and  the  steeper  it  is  elsewhere. 


Fig.  64. 


120  ELEMENTARY  FUNCTIONS 

0<n<l.  The  graph  is  tangent  to  the  2/-axis  at  the  origin, 
and  rises  as  it  runs  to  the  right.  The  smaller  the  value  of  n 
the  more  rapidly  it  rises  near  the  origin,  and  the  less  rapidly 
elsewhere. 

The  curves  of  these  two  groups  are  separated  by  the  graph 
of  y  =  X  {n  =  1);  the  graphs  for  two  reciprocal  values  of  n 
are  the  graphs  of  inverse  functions  and  are  symmetrical  to  each 
other  with  respect  to  this  Hne. 

n<0.  Both  axes  are  asymptotes  of  the  graph,  which  falls 
as  it  runs  to  the  right.  The  smaller  the  numerical  value  of  n, 
the  closer  the  graph  Ues  to  the  i/-axis,  between  x  =  0  and  a;  =  1, 
and  the  farther  it  Ues  from  the  a;-axis. 

The  curves  of  this  last  group  are  separated  from  those  of 
the  first  two  groups  by  the  line  y  =  1  (n  =  0),  the  graphs  for 
two  values  of  n  which  are  numerically  equal  but  differ  in  sign 
being  the  graphs  of  reciprocal  functions. 

For  three  values  of  n  in  any  one  of  these  three  groups,  the 
graph  corresponding  to  the  intermediate  value  of  n  always  Ues 
between  the  graphs  for  the  other  values  of  n. 

The  graph  of  ax"  may  be  obtained  from  that  of  a;"  by  the 
theorem  in  Section  30. 

EXERCISES 

1.  Find  the  reciprocal  of  each  of  the  following  functions,  and  sketcl 
the  graphs  of  both  functions: 

(a)  a^.  (b)  X*.  (c)  xK  (d)  xl  (e)  x*. 

2.  Find  the  inverse  and  the  reciprocal  of  each  of  the  functions  belowj 
and  sketch  the  graphs  of  the  three  functions  on  the  same  axes. 

(a)  x\  (b)  x^.  (c)  xi.  (d)x*. 

3.  On  the  same  axes,  plot  the  graph  of  the  function  indicated  below  fc 
the  given  values  of  a. 

(a)  ax^  for  a  =  1,  i,  |,  2,  3. 

(b)  axi  for  a  =  1,  ^,  2,  3. 

(c)  axi  for  a  =  1,  -  1,  -  i,  -  2. 

(d)  a/x  for  a  =  1,  4,  8,  -  1. 

(e)  a/x2  for  a  =  1,  2,  4,  6. 

4.  On  the  same  axes,  using  as  large  a  scale  aa  possible,  construct  the  j 
graph  of  x"  forn  =  -  2,  and  -  4.  What  can  be  said  of  the  form  of  the  i 
graph  if  n  is  an  even  negative  integer? 


I 


ALGEBRAIC  FUNCTIONS  121 


5.  On  the  same  axes,  using  as  large  a  scale  as  possible,  construct  the 
graph  of  a;'*  for  n  =  -  1  and  -  3.  What  can  be  said  of  the  graph  of  x« 
if  n  is  an  odd  negative  integer? 

6.  Construct  the  graph  of  x"  if  n  =  -  |  (a)  By  regarding  the  function 
as  the  reciprocal  of  the  function  xh;   (b)  By  regarding  it  as  the  inverse  of 

7.  Construct  the  graph  of  (a)  x  ~i,  (b)  x  "I,  in  each  case  using  the 
properties  of  the  graphs  of  reciprocal  functions.  What  relation  exists 
between  these  two  functions? 

8.  The  quantity  of  water  which  flows  from  an  exit  pipe  h  feet  below  the 
surface  of  a  reservoir  is  given  by  the  equation  q  =  -  S.02aVh,  where  a  is 
the  area  of  the  cross  section  of  the  pipe.  Assume  a  =  1,  and  plot  the  graph. 
If  one  outlet  pipe  is  10  feet  below  the  surface  and  another  is  twice  as  deep, 
is  the  flow  of  one  twice  that  of  the  other?     Why? 

r  —      9.  The  pressure  p,  the  volume  v,  and  the  temperature  t  of  a  gas  are 
■fcnnected  by  the  relation  pv  =  kt,  where  fc  is  a  constant.     Assuming  A;  =  1, 
^Hot  the  graph  if  (1)  p  is  constant,  (2)  v  is  constant,  (3)  t  is  constant. 
milO.   As  water  runs  out  of  a  basin,  a  depression  is  formed  near  the  outlet. 
^^^A  vertical  section  of  this  depression  in  the  surface  of  the  water  is  called  the 
curve  of  whirl.     It  is  given  by  the  equation  y  =>  -  hh-^/x^,  where  r  is  the 
radius  of  the  orifice  in  the  outlet  pipe,  and  h  is  the  head,  or  depth  of  water. 
The  head  changes  from  instant  to  instant,  so  that  the  curve  of  whirl  con- 
stantly changes.     Assuming  r  =  1,  plot  the  graph  for  /i  =  |,  1,  2. 

11.  Plot  the  graphs  of  the  functions  following,  and  then  sketch  the 
graphs  of  the  reciprocal  functions.     Find  the  reciprocal  function  in  each 


(a)  x-3. 

(b)  2-x. 

(c)    X^-1. 

(d)  x2  -  4. 

(e)  x/(x^  +  l). 

(f)   x2-7x  +  10. 

(g)  X'  -  4x. 

(h)  x^  -  4x. 

12.  What  is  a  simple  way  of  sketching  the  graph  of  l/(2a;  +  3)?  Of 
3/(2x  +  3)? 

42.  Interpolation.  The  process  of  finding,  for  example,  the 
cube  of  such  a  number  as  2.647,  which  lies  between  two  suc- 
cessive nmnbers  2.64  and  2.65  whose  cubes  are  given  in  the 
table,  is  called  interpolation.  According  to  Huntington^s 
Tables,  2.64^  =  18.40  and  2.65^  =  18.61.  The  difference  be- 
tween these  two  values  oi  y  =  x^  is  Ay  =  0.21,  and  is  called  the 
tabular  difference.  In  general,  the  tabular  difference  is  the 
difference  between  two  successive  niunbers  in  the  body  of  a 
table.    For  the  process  of  interpolation  it  is  essential  that  the 


122 


ELEMENTARY  FUNCTIONS 


successive  values  of  the  tabular  difference  should  be  very 
nearly  equal. 

The  table  gives  one  row  of  Huntington's  table  of  cubes  and 
the  successive  tabular  differences.  Since  Ay  changes  only 
shghtly  while  Ax  is  always  equal  to  0.01,  the  average  rate  of 
change,  Ay /Ax,  is  nearly  constant,  and  the  part  of  the  graph  of 
x^  constructed  from  this  table  would  be 
very  nearly  a  straight  Une.  Some  of  the 
successive  values  of  Ay  are  equal,  due  to 
the  fact  that  the  values  of  x^  are  approxi- 
mate to  four  figures  and  not  exact,  and 
hence  some  of  the  points  plotted  from  the 
table  would  actually  He  on  a  straight  line. 
For  the  process  of  interpolation  we  dssume 
that  the  part  of  the  graph  lying  between  two 
successive  points  is  straight. 

To  find  2.647^,  consider  that  part  of  the 
graph  lying  between  x  =  2.64  and  2.65,  as 
given  in  Figure  65.  We  seek  the  value  of 
the  ordinate  EF  at  the  point  E  for  which 
X  =  2.647.  It  may  be  obtained  by  adding 
GF  as  a  correction  to  AB  =  2M\  To 
find  GF,  we  have  the  slope  of  BF  is  the 
same  as  that  of  BD,  since  the  graph  is  assumed  straight.  Hence 


X 

2/=a:3 

Ay 

2.60 

17.58 

.20 

2.61 

17.78 

.20 

2.62 

17.98 

.21 

2.63 

18.19 

.21 

2.64 

18.40 

.21 

2.65 

18.61 

.21 

2.66 

18.82 

.21 

2.67 

19.03 

.22 

2.68 

19.25 

.22 

2.69 

19.47 

.21 

2.70 

19.68 

whence 


(?F  = 


GF[ 
BG 

0.007 


HD 
BH 


or 


GF 
0.007 


0.21 

o.or 


XO.21  =0.7x0.21  =0.147  =  0.15. 


I 


0.01 

Then  2.647^  =  EF  =  AB+GF  =  18.40  +  .15  =  18.55. 

EF  may  also  be  obtained  by  subtracting  ID  as  a  correction 
from  CD  =  2.653.  To  find  ID,  we  have  the  slope  of  FD  is 
equal  to  that  of  BD,  since  the  graph  is  assumed  straight,  so  that 
ID  ^HD  ID       0.21 

BH' 
0.003 


and  hence 


FI 
ID 


or 


0.01 


0.003 
xO.21 


0.01 

0.3x0.21  =0.06. 


ALGEBRAIC   FUNCTIONS 


123 


2.64 


2.647 


2.65 


FlQ.  66. 


Then  2.647^  =  EF  =  CD  -  ID  =  18.61  -  0.06  =  18.55,  as  be- 
fore. 

The  latter  procedure  is  usually  preferable  if  the  figure  for 
which  we  are  interpolating,  7  in  the  illustration,  is  greater  than 
5,  while  the  former  is  -^ 


used  if  it  is  less  than, 
or  equal  to,  5.  No- 
tice that  the  correc- 
tion GF  is  found  by 
taking  seven-tenths 
of  the  tabular  differ- 
ence, HD  =  0.21,  and 
the  correction  ID  by 
taking  three-tenths 
of  the  tabular  differ- 
ence. 

The  arithmetical  operations  involved  in  interpolation  may 
be  described  as  follows : 

To  find  2.64a^,  where  a  stands  for  the  digit  for  which  we  are 
interpolating,  apply  a-tenths  of  the  tabular  difference  as  a  cor- 
rection to  2.64^  if  a<5,  while  if  a>5,  apply  (10  —  a)-tenths  of 
the  tabular  difference  as  a  correction  to  2.65^.  In  either  case, 
the  correction  must  be  applied  (added  or  subtracted)  in  such  a 
way  that  the  result  lies  between  2.64^  and  2.65^. 

In  practice  the  decimal  point  is  usually  omitted  in  x,  x^,  and 
the  tabular  difference,  and  inserted  in  the  proper  place  at  the 
end  of  the  operation. 

Example  1.     Find  26.02'.     The  figure  for  which  we  must  interpolate 

is  2,  and   the   tabular  difference   is  20.      Two-tenths    of    the    tabular 

f  difference  is  4,  the  correction  to  be  applied  to  1758,  the  cube  of  260. 

"^  i  [n  order  to  obtain  a  result  between  1758  and   1778,  the  cube  of  261, 

i:  phe  correction  must  be  added,  giving  1762  as  the  first  four  digits  in 

ijl  ,:he  required  cube.    From  the  rule  at  the  top  of  the  table,  we  see  that 

J6.023  =  17,620. 

li  Example  2.     Find  2.687'.     Instead  of  applying  7-tenths  of  the  tabular 
jfJifference  as  a  correction  to  the  cube  of  2.68,  we  apply  3-tenths  of  it  to 
fhe  cube  of  2.69.    Three-tenths  of  the  tabular  difference  is  6.6  =  7,  and 
'    -lence  2.687'  =  19.40. 


124  ELEMENTARY  FUNCTIONS 

Except  for  purposes  of  explanation,  there  is  no  need  of  writing 
anything  but  the  desired  result. 

All  the  tables  in  Huntington's  Tables  are  arranged  so  that 
the  numbers  in  the  body  of  any  table  increase  as  we  read  from 
left  to  right,  for  the  reason  that  a  uniform  procedure  is  obtained 
for  applying  the  correction.  In  interpolating,  the  desired  re- 
sult lies  between  two  numbers  in  the  same  row;  a  correction 
applied  to  the  left-hand  number  is  always  added,  while  a  cor- 
rection to  be  applied  to  the  right-hand  ninnber  is  always 
subtracted. 

EXERCISES 

1.  Plot  the  graph  of  x^  from  x  =  4.50  to  x  =  4.60,  using  the  values  of 
x^  given  in  Huntington's  Tables. 

2.  If  we  plot  the  graph  of  Vx  for  values  of  x  from  2.30  to  3.40,  using 
the  square  roots  given  in  the  Tables,  why  will  the  graph  be  nearly  straight? 

3.  Find  the  values  of  the  numbers  given  below,  illustrating  the  inter- 
polation graphically. 

(a)  3.1722.       (b)  VOTS.       (c)  6.49^         (d)  ^^^0.02814.        (e)    1/926. 

4.  Find  the  numbers  following  without  illustrating  the  interpolation 
graphically.  The  tabular  difference  and  the  necessary  number  of  tenths  of 
that  difference  should  be  obtaified  mentally. 


„ 


(a)  1.5342,  V0.468,  8.433,  ^^^:46,  1/647. 

(b)  Squares  of  2.784,  3762,  0.01388,  3846000,  0.00003728. 

(c)  Square  roots  of  0.634,  3.248,  42.7,  384.3,  279,000,  0.001876. 

(d)  Cubes  of  3.143,  0.774,  1683,  0.00004592,  4889000. 

(e)  Cube  roots  of  0.02258,  0.226,  19.34,  0.00176,  328000. 

(f)  Reciprocals  of  31.76,  0.00647,  35990,  0.0004325,  647. 

6.   Find  the  square  of  4732,  and  check  the  result  by  finding  its  squ 
root.     Find  the  cube  root  of  3479,  and  check  the  result  by  cubing  it, 
Find  the  reciprocal  of  25.63,  and  check  the  result. 

6.  FindVSTTS^,   ^0.02478^,  1/73-262,  1/V2438,  1/48-363. 

7.  Solve  the  following  equations,  using  the  Tables  to  simplify  the  com 
putations: 

(a)  x^  +  32x  +  19-0.  (b)  2x^  -  A7x  -  27  =  0. 

(c)  3x2  ^  29x  -  40  =  0.  (d)  5x2  +  54x  -  31  =  0. 

8.  How  long  an  umbrella  will  go  into  a  trunk  measuring  31.5  x  18. 
X  22.5  inches,  inside  measure,  (1)  if  the  umbrella  is  laid  on  the  bottom! 
(2)  if  it  is  placed  diagonally  between  opposite  comers  of  the  top  an<! 
bottom?  ! 


ALGEBRAIC   FUNCTIONS  125 

9.  A  house  with  a  gambrel  roof  is  28.5  feet  wide,  the  first  set  of  rafters 
has  a  slope  of  |,  and  the  top  set  a  slope  of  f .  If  the  joint  in  the  roof  comee 
7.8  feet  from  the  side  of  the  building  measured  horizontally,  how  long  is 
each  set  of  rafters? 

10.  If  five  individuals  weigh  120,  124,  116, 112,  123  pounds,  respectively, 
and  five  others  95,  150,  132,  105,  113  pounds,  respectively,  then  the  aver- 
age weight  M  of  either  group  is  119.  But  one  group  is  distributed  very 
closely  around  the  mean  M,  whereas  the  other  group  exhibits  marked 
deviations  from  it.  A  measure  of  the  variability  or  tendency  to  devia- 
tion of  measurements  is  given  by  the  following  formula,  called  the  stand- 
ard deviation. 

-— —  The  average  M  is  found,  and  each  in- 

2  +  •  •  •  +    n     dividual  measurement  is  subtracted  alge- 


.D.  =  J*l± 


braically  from  M,  thus  obtaining  a  series 
of  deviations  d.  The  sum  of  the  squares  of  these  deviations  is  divided  by 
n,  the  number  of  measurements,  and  the  square  root  of  the  quotient  is  the 
standard  deviation  from  the  average  M. 

The  coefficient  of  variability  is  defined  as  the  ratio  of  the  standard 
deviation  from  the  average  to  the  average. 

Compare  the  coefficients  of  variability  for  the  two  sets  of  measurements. 

11.  The  strength  of  grip  of  right  hand  and  left  hand,  in  hectograms,  for 
10  boys  is  given  in  the  following  table.  Compute  the  standard  deviation 
and  coefiicient  of  variability  for  each. 

Right  hand.    158,    200,     210,    226,    248,    270,    296,    320,    348,    403. 
Left  hand.       138,     185,    200,    224,    244,    260,    282,    305,    336,    400. 

43.  Variation.  Definition.  It  is  said  that  y  varies  as,  or 
is  proportional  to,  the  nth  power  of  x  ii  y  =  fcx",  n  being  positive, 
wliile  y  varies  inversely  as,  or  is  inversely  proportional  to,  the  nth 
power  of  X  H  y  =  k/x"".  If  it  is  desired  to  contrast  these  two 
forms  of  variation,  the  former  is  called  direct  variation  as  op- 
posed to  inverse  variation. 

The  case  of  direct  variation  in  which  n  =  1  has  already  been 
considered  in  Section  21,  page  61.  In  any  case,  the  value  of 
the  constant  k  may  be  determined  from  a  given  pair  of  values 
of  x  and  y,  as  in  that  section,  by  substituting  the  given  values 
of  X  and  y. 

The  graph  of  the  relation  y  =  fcx"  may  be  obtained  from  one 
of  the  curves  considered  in  Sections  38  to  40,  by  means  of  the 
theorem  in  Section  30. 

The  language  of  variation  is  used  frequently  in  the  applica- 


126  ELEMENTARY  FUNCTIONS 

tions  of  mathematics.  Thus  for  bodies  moving  with  velocities 
near  that  of  a  rifle  ball,  the  resistance  of  the  air  varies  as  the 
cube  of  the  velocity.  The  intensity  of  hght  varies  inversely 
as  the  square  of  the  distance  from  the  source  of  Hght,  etc. 

The  language  of  variation  is  extended  also  to  apply  to  rela- 
tions involving  more  than  two  variables.  One  quantity  is 
said  to  vary  jointly  as  two  or  more  other  quantities,  if  the  first 
is  equal  to  a  constant  times  the  product  of  the  others:  direct 
and  inverse  variation  may  both  be  involved.  For  example 
Newton's  law  of  gravitation  states  that  the  attraction  of  two 
bodies  varies  jointly  as  their  masses  and  inversely  as  the  square 
of  the  distance  between  them.  If  the  masses  of  the  bodies  are 
m  and  m',  and  if  d  is  the  distance  between  them,  then  the  at- 
traction A  is  given  by  A  =  kmm'/(P. 


EXERCISES 

1.  If  the  area  of  a  rectangle  is  12  square  inches  the  altitude  varies  in- 
versely as  the  base.  Represent  the  relation  graphically.  On  the  figure 
construct  several  rectangles  of  area  12. 

2.  A  man  told  a  contractor  that  he  wished  to  have  certain  work  done  in 
the  next  three  days.  The  contractor  replied  that  it  would  be  impossible, 
as  it  would  take  six  men  three  weeks  to  do  it.  Whereupon  the  man  told 
him  that  thirty-six  men  would  do  it  in  three  days,  and  to  have  them  on 
the  job  the  next  morning.  What  relation  did  he  assume  exists  between  the 
number  of  men  and  the  number  of  days  in  which  they  could  do  the 
work? 

3.  An  assumption  which  a  psychologist  has  made  states  that  the  law- 
lessness of  a  mob  varies  as  the  square  of  the  number  of  individuals  in- 
volved. Illustrate  by  a  graph.  Compare  the  lawlessness  of  two  mobs 
of  50  men  and  500  men. 

4.  The  mean  distance  of  the  earth  from  the  sun  is  93,000,000  miles  and 
from  the  moon  is  240,000.  If  the  mass  of  the  earth  is  taken  as  1,  the 
masses  of  the  sun  and  moon  are  approximately  330,000  and  1/81 
respectively. 

(a)  Does  the  sun  or  the  earth  exert  the  greater  attraction  on  the  moon? 

(b)  Does  the  sun  or  the  moon  exert  the  greater  attraction  on  the  earth? 

(c)  At  what  distance  from  the  earth  would  a  particle  be  equally  attracted 
by  the  earth  and  the  moon?     By  the  earth  and  the  sun? 

5.  The  horse  power  required  to  propel  a  boat  varies  approximately  as 
the  cube  of  its  velocity.     If  a  58  H.P.  engine  will  produce  a  velocity  of 


ALGEBRAIC  FUNCTIONS  127 

10  feet  per  second,  what  horse  power  will  produce  a  velocity  of  20  feet 
per  second?  Plot  the  graph  of  the  relation  and  illustrate  the  above  data 
on  it. 

6.  The  weight  of  a  metal  disk  of  given  thickness  and  material  varies 
as  the  square  of  the  diameter,  (a)  If  a  disk  whose  diameter  is  1  inch 
weighs  ^  ounce,  what  is  the  weight  of  a  disk  whose  diameter  is  4  inches? 
What  is  the  diameter  of  a  disk  weighing  7  ounces?  Sketch  the  graph 
showing  the  weight  of  any  such  disk,  (b)  If  a  given  disk  weighs  12  ounces, 
how  would  its  diameter  compare  with  one  whose  weight  is  6  ounces? 

7.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diameter.  How 
many  shot  i  of  an  inch  in  diameter  can  be  made  by  melting  a  lead  sphere 
whose  diameter  is  2  inches? 

8.  The  intensity  of  light  varies  inversely  as  the  square  of  the  distance 
from  the  source  of  light.     If  an  object  is  20  feet  from  a  light,  by  how  much 

lust  it  be  moved  to  receive  twice  as  much  light?     Illustrate  graphically. 

9.  If  an  object  is  3  feet  from  a  light,  how  far  should  it  be  moved  to  re- 
ive one-third  as  much  light? 

10.  The  volume  of  a  cylinder  varies  jointly  as  the  altitude  and  the 
|uare  of  the  diameter.     A  preserving  kettle  12  inches  in  diameter  is  filled 

a  depth  of  8  inches.     How  many  quart  jars,  3|  inches  in  diameter  and 
inches  high,  will  be  needed  to  hold  the  preserves  in  the  kettle? 

11.  In  enlarging  a  photograph,  the  original  negative  is  projected  on  a 
3itized  plate,  just  as  a  lantern  shde  is  projected  on  a  screen.     The  size 

rea)  of  the  enlargement  varies  as  the  square  of  the  distance  from  the 
source  of  light,  and  is  equal  to  the  original  if  the  sensitized  plate  is  placed 
right  against  the  negative.  If  a  4  x  5  negative  is  placed  ten  inches  from 
the  source  of  light,  where  should  the  sensitized  plate  be  placed  to  obtain 
an  enlargement  8  x  10  inches? 

44.  Empirical  Data  Problems.  Since  the  equation  y  =  kx"" 
may  be  written  in  the  form  ?//x"  =  k,  the  table  of  values  of  x 
and  y  is  such  that  the  quotients  obtained  by  dividing  the  valites  of 
y  by  the  nth  powers  of  the  corresponding  values  of  x  are  equal. 
This  property  enables  us  to  determine  whether  a  given  table 
of  values,  obtained  by  an  experiment,  may  be  represented  ap- 
proximately hy  y  =  kx"",  and  to  determine  the  coefficient  k. 
If  the  points  whose  coordinates  are  the  pairs  of  values  in 
the  table  are  plotted,  and  appear  to  lie  on  the  graph  of  A;x",  we 
'Will  have  n>l  if  the  graph  is  tangent  to  the  a:-axis  at  the 
origin,  0<n<l  if  the  graph  is  tangent  to  the  y-axis  at  the 
origin,  and  n<0  if  the  graph  approaches  the  axes  asymptoti- 
cally (Sunmiary,  Section  41). 


128  ELEMENTARY  FUNCTIONS 

If  n>l,  a  simple  value  to  choose  for  n  is  2.  By  the  above 
property,  if  the  quotients  obtained  by  dividing  the  values  of 
y  by  the  squares  of  the  corresponding  values  of  x  differ  by  very 
little,  the  relation  connecting  x  and  y  resembles  y  =  kx^^  and  an 
approximate  statement  of  the  law  may  be  given  in  this  form. 
Each  quotient,  y/x^,  is  the  value  of  k  for  which  the  graph  of 
y  =  kx^  passes  through  the  point  (x,  y).  A  very  good  value 
to  use  for  k  is  obtained  by  finding  the  average  of  all  the  quo- 
tients. If,  however,  the  quotients  differ  widely,  it  is  well  to 
try  n  =  3,  or  n  =  I,  or  some  other  value  of  n,  in  order  to  see 
if  an  approximate  statement  of  the  law  can  be  found  which  is 
better  than  y  =  kx^. 

If  0<n<l,  a  simple  value  to  choose  for  n  is  J,  but  if  the  re- 
sult is  not  satisfactory,  other  values  of  n  should  be  tried. 

If  n<0,  the  values  occurring  most  frequently  are  n  =  —  1 
and  n  =  -  2.  These  cases  may  be  distinguished,  provided 
that  the  same  unit  is  used  on  both  axes,  by  the  fact  that  the 
graph  for  n  =  —  1  is  symmetrical  with  respect  to  the  bisector 
of  the  first  and  third  quadrants.  If  neither  of  these  values  of 
n  proves  satisfactory,  others  should  be  tried.  If  n  is  negative , 
division  hy  a:"  is  equivalent  to  multiplication  by  x~9.  For  ex- 
ample, if  n  =  —  2,  we  have  y/x~^  =  yx^.  The  value  of  k  to 
be  chosen  is  now  the  average  of  certain  products  instead  of 
quotients. 

The  accuracy  with  which  the  equation  finally  obtained  repre- 
sents the  law  may  be  checked  as  on  pages  80,  83,  and  106. 

The  use  of  the  tables  of  squares,  square  roots,  etc.,  is  ad- 
vantageous in  finding  k. 

In  a  later  chapter  we  shall  obtain  a  more  satisfactory  method 
of  determining  n. 

EXERCISES 

1.  In  the  following  table  the  relation  between  x  and  yiay  =  k/x.  Con- 
i  1  ty  o  A  struct  the  graph  and  determine  k.  Hint.  Find 
'     '       '        *  '      the  product  of  x  and  y  for  each  pair  of  values 


t/4    21    15    09 

'     '   '     '   '     *   '     and  take  the  average  of  these  products.     Check 

the   accuracy  of  the  result. 


ALGEBRAIC   FUNCTIONS  129 

2.  In  the  following  table  the  relation  between  x  and  y  v&  y  =  k/x^. 
1,     2,        3,         4,  Plot  the  graph  and  determine  k.    Check  the 

3,  0. 78,  0. 32,  0. 19,     accuracy  of  the  result. 

3.  In  the  table  V  represents  the  volume  of  a  vessel  containing  a  gas 
I  0.1,  0.2,  1,  2,  10,  and  p  the  pressure  on  the  walls. 
I  10,     5.01,  0.99,  0.52,  0.13,     Find  the  relation  between  V  and  p. 

4.  Determine  the  law  of  the  attraction  A  of  an  electrified  rod  for  a 
0.51,  0.98,  1.53,  1.97,  2.56,  "^^^^f,  ^^  a  distance  d  as  given  in 
5.98,  1.48,  0.66,  0.38,  0.23,    ^^^  table.     What  is  the  attraction 

when  the  distance  is  1.2? 
6.   In  the  following  table  W  represents  the  effect  of  the  blow  when  a 
Q      J       2      3         4        weight  of  16  pounds  is  dropped  and  strikes 
o'  0  23    l'  2  37   4  01     ^^^^   ^   velocity   v.    Plot    the   data,   de- 
'*''*'*'    termine  the  law,  and  find  W  li  v  =  2.5. 

6.  The  discharge  of  water  through  a  large  pipe  line  or  conduit  is  meas- 

0,     1.00,       1.98,      2.98,       3.98,       ""'^^^^  ^  ^^'f^'lT^^''  °" 

0,  0.0442,  0.0637,  0.0780,  0.0909,     f^^t  "^  "''^^  *'"'  ^'f'^""^  "> 

height  of  a  column  of  mercury 

in  the  two  sides  of  a  [/-tube.     In  the  table,  x  denotes  the  difference  in 

height  and  Q  the  discharge.     Determine  the  law. 

7.  From  the  following  data  determine  the  resistance  R,  in  ohms  per 
d  I  0.083,  0.120,  0.148,  0.220,  mile,  of  a  telegraph  wire  of  diameter 
r\      51,24.42,     16.1,     7.26,      d,  in  inches. 

8.  Find  the  relation  between  the  pressure  p  and  the  volume  F  of  a  gas 
T/  I  K  AQ    o  Kft    1  oi    A  on    A  Ao      ^^  ^^^  pressure  on  the  walls  of  a  con- 

—    n  Ki    A  n^    o  A^    Q  AQ    Q  aA      taming  vessel  for  different  volumes 
V   I  0.51,  0.97,  2.04,  3.03,  3.94,  -       j  .     u         •    ii.    .  1 1 

was  found  to  be  as  m  the  table. 

9.  In  the  following  table,  x  denotes  half  the  distance  between  two  tele- 

X  I    20,       30,       40.       50,       60,       ^^"^^  P"'"''  f^^  ^  *';'=  ^°'^"'''  ^^ 

irlo.98,  2.21,  3.95,  6.19,  8.97,    T'^  ^'  ^*  ^^^  •'"°'"'-     ^""^  *'^« 
'  law. 

10.   The  table  gives  the  number  of  revolutions  per  minute,  R.P.M.,  at 

h       I  11.63,  62.8,  138.8,  174.5,     "!;'*   ^  ''<^'*1°  ^°™  f  7f 
'  wheel  runs  without  any  load,  for 


R.P.M.  I     440,     1020,    1525,     1680,      ,.«:        ^       ,  r    ^u      u     j     i, 

different  values  of    the    head,  h 

(the  height  of  the  surface  of  the  mill  pond  above  the  bottom  of  the  wheel). 

Find  the  relation. 

11.  The  horse  power  developed  by  the  water  wheel  in  Exercise  10  is 

Ji I  69.8,  104.8,,  139.8,    174.5,     given  in  the  table.     Find  the  law 

H.P.  I  0.64,     1.18,     1.81,     2.56,      in  the  form  i/ =  fcxi. 

12.  (a)  Economists  define  supply  as  the  amount  of  anything  which  the 

r>  •       I  A  ^^  1  ^n  -I    ^r.    i  on     producers  will  offer  for  sale  at  a  given 

Price      0. 60,  1 .  00,  1 .  40,  1 .  80,     *^  .  rru    .^  ui      •  j     i. 

o p-  — :r-^ — ^ — 5—^ — J—     price.     The  table  gives  an  assumed  set 

'         '         *  *       of  values  of  the  supply  of  wheat,  in  mil- 


130  ELEMENTARY  FUNCTIONS 

lion  bushels,  at  various  prices  in  dollars.     Plot  the  graph  and  determine 

the  law.     What  price  per  bushel  will  stimulate  the  producers  to  furnish  5 

miUion  bushels? 

(b)  The  term  demand  is  used  to  denote  the  quantity  of  anything  which 

T>  •  1  n  ^r^    A  on    1  f^f>    o  AA     ^^^  consumers  will  purchase  at  a 

Price         0.40,  0.80,  1.00,  2.00,       .  .         ^r       j  ^i      j  j 

'  given  pnce.     How  does  the  demand 


fluctuate  as  the  price  rises?  As  the 
price  falls?  The  table  gives  an  assumed  set  of  values  of  the  demand  for 
wheat,  in  million  bushels,  at  various  prices,  in  dollars.  Plot  the  graph 
and  determine  the  law.  What  price  will  stimulate  the  consumers  to  de- 
mand 7  miUion  bushels? 

(c)  Assuming  the  conditions  in  the  preceding  parts  of  this  exercise, 
plot  both  graphs  on  the  same  axes,  and  find  the  price  and  the  quantity 
of  wheat  transferred  from  producer  to  consumer. 

13.  If  the  apple  crop  amounts  to  3  million  bushels,  then,  at  the  harvest 

Price      I  0.50,     1,    1.25,2.50,     ''^°°'  *"  ™PP'>;  '**  ^7,  P"'=<'  ^'f 

K5S5^     5,   2.5,   2, IT    r"f  K?  ^"^  T  T"  "/• 

ihe  table  gives  assumed  values  for 
the  demand  in  million  bushels.  Plot  both  graphs  on  the  same  axes,  and 
determine  the  equations  giving  the  laws.  At  what  price  will  apples  sell 
under  these  conditions? 

14.  Some  industries  develop   approximately  to   a  point  where  any 

amount,  within  certain  limits,  can  be  supplied  at  a  constant  price.     Sup- 

T,  .         I  o    ..    tr    ^    o    1A      pose  that  any  number  of  shoes  will  be 
Price        3,  4,  5,  6,  8,  10,     ^       ,.    ,     ,   ,^        .        ,  ^_  .  , 

^Pi j-\     '   -     .    „      ' — z-^    supphed  at  the  price  of  $5  per  pair,  and 

Demand      6,  5,  4,  3,  2,    1,      ,,  ^f^  xi,     j  5    •         -n-  •      ' 

that  the  demand,  m  milhon  pairs,  at  a 

price,  in  dollars,  is  as  given  in  the  table.    Plot  both  graphs  on  the  same 

axes,  and  find  the  price  and  the  number  of  shoes  sold.     Why  cannot  the 

price  be  less  than  $5?     Assuming  competition,  why  will  it  not  exceed  $5? 

(b)  If  a  monopoly  arose,  with  the  object  of  making  as  much  money  as 
possible  instead  of  serving  the  public  to  the  best  of  its  ability  with  reason- 
able profits,  the  output  would  be  restricted  to  the  point  of  maximum  profits. 
Assuming  the  conditions  of  the  preceding  part  of  the  problem,  suppose 
that  the  cost  of  manufacturing  a  pair  of  shoes  is  $4.  Construct  a  table 
giving  the  total  profits  at  various  prices,  plot  the  graph,  and  determine  the 
price  and  profits. 

16.  Plot  the  graphs  and  determine  the  laws  for  the  data  given  in  the 
tables.     Determine  the  freight  rate  and  the  amount  of  freight  handled. 

Price  per  ton  mile  in  cents I  0.5,     Q.6,     0.7,     0.8, 

Supply  of  freight  service  in  thousand  tons        14,         3,        2,         1, 

Price  per  ton  mile  in  cents I  0.5,     0.6.     0.7,       1, 

Demand  for  freight  service  in  thousand  tons      |    3,      2.6,    2.1,     1.5, 


ALGEBRAIC   FUNCTIONS 


131 


45.  The  Linear  Fractional  Fxmction 


ax  +  b 


In   order   that 


V 

1 

l\ 

l^ 

\V 

yfe 

1      1 
-/     -3    -i 

-1 

/ 

N. 

?i-| 

«. 

j;;; 

-^ 

/ 

6  i 

^1    ' 

li 

X 

•^ 

' 

/ 

^ 

-J 

_J 

Fig.  66. 


cx  +  d 

this  function  should  really  be  fractional,  it  is  essential  that 

c  7^  0,  for  if  c  =  0,  it  reduces  to  the  linear  fiuiction  "j  ^  +  j 

The  simplest  Unear  fractional  function  is  \/x,  obtained  by- 
setting  6  =  c,  and  a  =  rf  =  0,  whose  graph  has  been  considered 
in  Section  40.  If  6  ?^  c,  while  a  =  c?  =  0, 
we  have  the  function  h/cx,  or  fc/x, 
where  k  =  h/c.  Its  graph  may  be  ob- 
tained by  multiplying  by  k  the  ordi- 
nates  of  points  on  the  graph  of  \/x 
(Theorem,  page  89). 

If  we  set  y  =  k/x^  whence  xy  =  k,  it 
is  seen  that  the  graph  is  symmetrical 
with  respect  to  the  line  y  =  x,  since 
the  equation  is  unchanged  if  x  and  y 
are  interchanged.  The  graph  is  also  synametrical  with  respect 
to  the  origin. 

The  geometric  significance  of  the  constant  A;  is  obtained  by 
solving  the  equations  xy  =  k  and  y  =  x;  the  solutions  are 
(V/c,  Vk)  and  (-  Vk,  -  Vk),  which  are  the  coordinates  of 
the  points  of  intersection  of  the  graphs  of  the  equations.  The 
distance  from  the  origin  to  the  point  of  intersection  A  is  found 
from  the  right  triangle  OAB  to  be  V2k.  Hence,  for  a  small 
value  of  k,  the  graph  lies  close  to  the  origin  and  axes,  while 
if  k  is  large,  it  lies  at  some  distance  from  them.  If  A;  is 
negative,  the  graph  will  lie  in  the  second  and  fourth 
quadrants. 

The  graph  oi  xy  =  k  is  called  a  rectangular,  or  equilateral, 
hyperbola.  The  former  term  is  derived  from  the  fact  that  the 
asjrmptotes  are  perpendicular. 

In  order  to  determine  the  form  of  the  graph  of  the  general 
Hnear  fractional  function,  set 


ax  +  h 
ex  +  d' 


(1) 


132  ELEMENTARY  FUNCTIONS 

Multiplying  both  sides  by  ex  +  b,  and  subtracting  ax  +  b 
from  both  sides, 

cxy  -  ax  +  dy  -  b  =  0.  (2) 

Let  us  now  see  if  we  can  simpUfy  this  equation  by  translat- 
ing the  axes  (Section  31,  page  89).    Setting 

X  ==  x'  +  h        and        y  =  y'  +  k,  (3) 

we  get 

cix'  +  h)  {y'  +  k)  -  a(x'  +  h)+  d{y'  +  k)  -  b  =  0,      (4) 

or,  removing  the  parentheses  and  collecting  like  terms, 

cxY  +  {ck  -  a)x'  +  {ch  +  d)y'  +  chk  -  ah  +  dk  -  b  =  0.  (5) 
Equating  to  zero  the  coefficients  of  x'  and  y\  we  get 

ck  -  a  =  0        and        ch  +  d  =  0.  (6) 

Solving  for  h  and  k,  which  is  always  possible  since  c  ?^  0, 

h  =  -  d/c,  k  =  a/c.  (7) 

Substituting  these  values  in  (5)  we  get 


ex 


v-(-9©-«(-^)-^'^^^=o. 


Simplifying,  subtracting  the  constant  term  from  both '  sides 
and  dividing  by  c, 

,  ,      be  —  ad  ,„. 

^y  =  — ^--  (8) 

This  equation,  referred  to  the  new  axes,  has  the  same  graph 
as  (1).  It  is  of  the  form  x'y'  =  k,  where  fc  =  (6c  -  ad)/e^,  or 
y'  =  k/x'.    The  graph  is  therefore  a  rectangular  hyperbola. 

Hence  we  have  the 

Theorem.  The  graph  of  a  linear  fractional  function  (1),  or 
of  an  equation  of  the  form  (2),  is  a  rectangular  hyperbola  whose 
asymptotes  are  parallel  to  the  axes  of  coordinates, 

EXERCISES 

1.  Plot  the  graph  of  ?/  =  {2x  -  4)/(a;  +  3),  (a)  by  translating  the  axes; 
(b)  by  finding  the  asymptotes  (by  the  method  on  page  27),  plotting  one 
branch  of  the  curve  from  a  table  of  values,  and  the  other  by  means  of  the 
symmetry  with  respect  to  the  point  of  intersection  of  the  asymptotes. 


ALGEBRAIC   FUNCTIONS  133 

2.  Plot  the  graphs  of  the  following  equations: 

a;  +  4  .,  ^  3x  +  6  ,  .  x  -  4 

(a)   2/  =  ^35-  (b)  2/  =  ^-3^-  (0  y  =  ^^^g- 

(d)  x!/  -  2x  +  42/  -  8  =  0.  (e)  2a;y  -  X  +  3?/  +  8  =  0. 

3.  The  linear  fractional  function  may  be  written  in  the  form 
a{x-{-'b/a)/c{x-\-  d/c).  Prove  that  the  linear  factors  cancel,  and  the 
function  is  a  constant,  if  and  only  if  be  -  ad  =  0.  For  this  reason,  it  is 
always  assumed  that  he  -  ad  5^  0. 

4.  The  relation  between  the  radius  of  curvature  of  a  concave  mirror, 
R,  the  distance  from  the  center  of  the  mirror  to  the  object,  x,  and  the  dis- 
tance from  the  center  to  the  reflected  image,  y,  is 

211 

ft  R    i^y 

If  JR  =  2  feet,  construct  the  graph  of  the  equation. 

46.  Integral  Rational  Functions.  The  general  form  of  an 
integral  rational  function  or  polynomial  is  (see  page  39) 

P^  Six)  =  oox"  +  aia;"-i  +  .  .  .  -H  an-ix  +  a„, 

where  n  is  an  integer,  and  ao,  ai,  .  .  .  a„  are  constants,  posi- 
tive, negative  or  zero,  except  that  ao  7^  0.     It  is  said  to  be  of 
•  degree  n. 
We  have  already  studied  polynomials  of  the  first  and  second 
degree  in  considering  linear  and  quadratic  functions. 

The  calculation  of  a  table  of  values  for  a  polynomial  of 
higher  degree  than  the  second  by  direct  substitution  presents 
no  new  features,  but  because  of  the  greater  number  of  terms 
that  may  be  present  there  is  more  labor  involved.  An  alterna- 
tive method  which  is  simpler,  in  general,  than  that  of  direct 
substitution  is  developed  in  the  next  sections. 

47.  The  Remainder  Theorem. 

Example.  Let  us  divide  the  polynomial  2x^  -6x^  +  11  by  x  -  2,  ar- 
ranging the  work  as  usual, 

2rr3  -  6x2  ^  H  \x  -2 

2a^-4a;2  |2x2  -  2x  -  4    ' 

-2x2  . 

-  2x2  ^  4a; 
—  4x 
-4x  +  8 

+  3 


134  ELEMENTARY  FUNCTIONS 

The  result  may  be  put  in  the  form 

^-^^^±11  =  2.-2.-4.^,  (X) 

where  2a:2  -  2a:  -  4  is  the  quotient,  denoted  by  q{x),  and  the  remainder  is 

Equation  (1)  is  an  identity  in  accordance  with  the 
Definition.    An  identity  is  an  equation  which  is  satisfied  by 
all  values  of  the  variable  or  variables. 

Substituting  2  for  a;  in  the  polynomial  in  the  example, 

/(2)  =  2.23  _  6  22  +  11  =  16  -  24  +  11  =  3. 

Hence  the  remainder,  i2  =  3,  obtained  by  dividing  the  poly- 
nomial by  x  —  2  is  equal  to  the  value  of  the  polynomial  when 
2  is  substituted  for  x. 

This  result  is  a  verification  of  a  theorem  which  is  proved 
for  any  polynomial  as  follows : 

Remainder  Theorem.  If  a  polynomial  f(x)  is  divided  hy 
X  —  a,  the  remainder  is  f{a) . 

If  f{x)  be  divided  by  x  —  a,  the  quotient  q{x)  is  a  poly- 
nomial of  degree  one  less  than  that  oi  f(x),  and  the  remainder 
22  is  a  constant. 

The  result  of  the  division  may  be  expressed  in  the  form 

X  -  a                  X  -  a  ^ 
Multiplying  both  members  by  a:  -  a,  we  obtain  the  identity 

fix)  =  (x-a)  q(x)  +  R.  (2) 
Substituting  a  for  x,  we  have 

f{a)  =  (a  -  a)q{a)  +  R.  (3) 
Since  a  -  a  =  0,  we  have  (a  -  a)q{a)  =  0. 

Hence                       /(a)  =  R.  (4) 

48.  Synthetic  Division.  The  form  of  the  division  in  the 
example  of  the  preceding  section  is  too  cumbersome  to  be  of 
practical  use  in  calculating  the  table  of  values  of  a  polynomial. 
The  division  may  be  simplified  as  follows: 


ALGEBRAIC   FUNCTIONS 


135 


We  note  that  the  operations  involved  in  the  division  were 
performed  on  the  coefficients  of  the  dividend,  divisor  and 
quotient,  the  powers  of  x  serving  merely  to  determine  the  proper 
positions  of  these  coefficients  in  carrying  out  the  details  of  the 
division.  If  missing  terms  are  suppHed  by  zero  coefficients, 
the  division  can  be  performed  by  the  so-called  method  of  de- 
tached coefficients  as  follows: 


2-6  +  0+11 
2-4 

-2 
-2  +  4 


1 


2-2-4 


-4 

-4+  8 
+  3 
The  first  term  in  each  partial  product  is  canceled  in  the 
ibtraction  and  hence  may  be  omitted.     The  second  term  of 
Lch  partial  product  may  be  written  immediately  under  the 
[term  in  the  dividend  from  which  it  is  subtracted.    The  first 
jrm  in  the  divisor  is  not  now  needed,  so  that  the  process  is 
iher  condensed  as  follows: 

2*  -  6    +0    +11  I   -2 

-4+4-8 


2-2-4 


_  2*  -  4*  +    3 


Noting  that   the  coefficients  of  the   quotient  are   identical 

ith  the  numbers  marked  with  an  asterisk  (*),  it  is  unnecessary 

write  them  under  the  divisor.    As  we  shall  use  this  process 

to  find  the  value  of  /(2),  it  is  convenient  to  replace  —  2  in  the 

divisor  by   +  2  and  add  throughout  instead  of  subtracting. 

Also  bringing  down  the  first  coefficient  of  the  dividend,  the 

work  is  now  arranged  in  the  form 

2-6  +  0  +  11  I  2 

+4-4-    8 
2-2-4+    3 

The  numbers  in  the  last  row  are  called  partial  remainders, 
le  last  being  the  remainder  3  (the  value   of  the  polynomial 


136  ELEMENTARY  FUNCTIONS 

when  2  is  substituted  for  x).  The  preceding  partial  remainders 
2,  -  2,  -  4  are  the  coefficients  of  the  quotient,  the  first  being 
the  same  as  the  first  coefficient  of  the  dividend. 

The  process  of  synthetic  division  may  be  described  as  follows: 

To  divide  a  polynomial  by  x  —  a, 

Write  the  coefficients  in  the  order  of  descending  powers  of  x, 
supplying  missing  terms  with  zero  coefficients.  To  the  right 
write  a. 

Bring  down  the  first  coefficient  of  the  dividend. 

Multiply  the  first  coefficient  by  a,  and  add  the  product  to  the 
second  coefficient. 

Multiply  the  sum  so  obtained  by  a,  and  add  the  result  to  the 
third  coefficient,  etc. 

The  last  term  will  be  the  remainder,  R,  and  the  preceding 
partial  remainders  will  be  the  coefficients  of  the  quotient,  q{x), 
which  will  be  of  one  degree  less  than  the  degree  of  the  polynomial. 

The  remainder  theorem,  with  the  aid  of  synthetic  division, 
affords  a  simple  method  of  calculating  the  value  of  a  poly- 
nomial for  a  given  value  of  x. 

For    example,    if    fix)  =  3x*  -  9x^  -  4^2  -  17a;  -  37,    find 
/(4).     By  the  remainder  theorem,  /(4)     3_  9_  4_i7_37i4 
is  the  value  of  the  remainder,  R,  when        +  12  +  12  +  32  +  60 
fix)  is  divided  by  x  -  4.    By  synthetic     3+3+  8  +  15 +"23 
division  we  find  that  /(4)  =  23.     Verify  this  result  by  substitut- 
ing 4  for  a;  in  the  given  polynomial. 

If  the  multiplications  and  additions  are  performed  mentally, 
the  partial  products  in  the  second  row  of  numbers  omitted, 
and  the  partial  remainders  alone  put  down,  the  computation 
may  be  arranged  still  more  compactly  in  the  following  form: 

The  second  arrangement  is  more 
convenient  in  calculating  a  table  of 
values  of  a  polynomial  for  integral 
values  of  x,  while  we  shall  find  that  the  first  is  better  adapted 
to  finding  zeros  of  the  function  if  the  zeros  are  irrational. 

49.  Graph  of  a  Pol3momial.  Example.  Construct  the 
graph  of 

fix)  =  x3  -  3x2  _  10a;  +  24. 


3_9_4-17-37 

3+3+8  +  15+23 


ALGEBRAIC   FUNCTIONS 


137 


The  table  of  values  is  constructed  by  means  of  synthetic 
division  and  the  remainder  theorem,  the  details  of  the  work 
and  the  graph  being  given  below. 


1  _  3  _  10  +  24 

+  1  -    2-12 

1-2-12  +  12 


1 


1  -  3  -  10  4-  24  f  2 

■1.2-    2-24 
1  _  1  _  12  +    0 


Hence  we  find  that /(I)  =  12,  and/(2)  =  0. 

The  calculations  may  be  arranged  compactly  by  writing  the 
coefficients  of  the  polynomial  once  for  all  as  in  the  top  row  of 
the  following  table  and  by  operating  on  this  top  row  with  each 
value  of  X,  performing  the  multiplications  and  additions  men- 
tally and  entering  the  partial  remainders  on  the  same  row  with 
the  value  of  x,  under  the  proper  coefficient  of  the  dividend. 


X 

fix) 

0 

1-3-10 

24 

1 

-2-12 

12 

2 

-  1  -  12 

0 

3 

0-10 

-    6 

4 

+  1-6 

0 

5 

+  2        0 

24 

6 

+  3+    8 

72 

-1 

-4-    6 

30 

-2 

-5        0 

24 

-3 

-6+    8 

0 

-4 

-7  +  18 

-48 

Fig.  67. 


60.  Extent  of  the  Tables.  In  the  example  of  the  preceding 
section  an  examination  of  the  synthetic  division  by  x  —  6 
shows  that  the  successive  remainders  1,  +3,  +8,  +72  are  aU 
positive.  If  a  larger  value  of  x  be  used,  by  the  nature  of  syn- 
thetic division,  it  is  evident  that  the  successive  remainders 
will  be  positive  and  larger.  Hence  as  x  increases  beyond  6 
f{x)  will  also  increase  and  the  curve  will  run  up  to  the  right 
indefinitely.  The  synthetic  division  by  a;  +  4  gave  remainders 
1,   -  7,  +  18,   -  48  which  have  alternating  signs.    As  x  de- 


138  ELEMENTARY  FUNCTIONS 

creases  below  -  4,  f{x)  also  decreases  and  hence  the  curve  runs 
down  indefinitely  to  the  left. 

The  extreme  values  of  x  that  should  be  included  in  the  table 
are  determined  by  the  following  rule: 

The  largest  value  included  in  the  table  is  such  that  in  the  syn- 
thetic division  the  signs  of  the  partial  remainders  are  all  the  same. 

The  smallest  value  {algebraically)  included  is  such  that  these 
signs  alternate. 

EXERCISES 

1.  Plot  the  graphs  of  each  of  the  polynomials  below.  From  the  graph 
find  approximate  values  of  the  roots  of  the  equation  obtained  by  equating 
the  polynomial  to  zero. 

(a)    x2-2x  +  7.  (b)  X*  -  3x3  +  2x2.        (c)  x3^3a.2  _  jq^.  _  24. 

(d)  2x3  -  8x2  -  12a;  +  17.  (e)  x^  +  3x2  +  3^;  +  1.  (f)  x^  -Sx*-x  +  3. 

2.  The  graphs  of  the  functions  x^  -  x2,  2x^  -  3x2  ^  ^^  2x^  -  x"^  -  x  pass, 
through  the  origin  and  the  point  (1,  0).  Find  the  slope  of  the  tangent  line 
to  each  graph  at  these  points,  and  plot  the  graphs.  How  does  the  slope  of 
the  tangent  line  assist  in  drawing  the  graphs? 

3.  Plot  the  graph  of  x^  -  4x3  4-  4^2^  ^n^j  ghow  that  it  is  tangent  to  the 
X-axis  at  two  points. 

4.  A  body  moves  so  that  its  position,  s,  with  reference  to  a  fixed  point 
at  any  time  t,  is  given  by  one  of  the  following  equations.  Find  the  velocity, 
V,  and  the  acceleration,  a,  at  any  time  t.  Construct  the  graphs  of  s,  v, 
and  a  on  the  same  set  of  axes.  When  and  where  is  the  body  at  rest? 
What  is  the  value  of  the  acceleration  when  the  velocity  is  a  minimum? 
Where  is  the  body  at  this  moment  and  what  is  its  velocity? 

(a)  s  =  f3  -  Zt.  (b)  5  =  <3  4-  3^2. 

(c)  s  =  <3  _  6^2  +  9f.  (d)  s  =  <3  +  3<2  _  ^  _3^ 

Suggestion.  To  find  the  acceleration,  a,  at  any  time  t,  find  the  limit  of 
the  average  rate  of  change  of  v  with  respect  to  t. 

Note.  Rate  of  change  of  a  polynomial.  It  was  seen  on 
page  111  that  the  rate  of  change  of  x"  is  nx""-^,  and  it  may  be 
shown  that  the  rate  of  change  of  ax''  is  anx""'^.  Thus  the  rate 
of  change  of  ^  is  12.^2,  and  that  of  -  2x*  is  -  Sx^.  It  will  be 
show  in  a  later  chapter  that  the  rate  of  change  of  a  pol5aiomial, 
or  the  slope  of  the  line  tangent  to  the  graph,  may  be  obtained 
by  adding  the  rates  of  change  of  the  successive  terms,  noting 
that  the  rate  of  change  of  the  constant  term  is  zero. 


ALGEBRAIC  FUNCTIONS  139 

For  example,  the  rate  of  change  m  of  the  polynomial 
2/  =  ^  -  3a;^  +  4a;2  -  5a;  +  7 
is  7W  =  4x3  _  9^2  _|.  83.  _  5^ 

6.  Find  the  slope  m  of  a  line  tangent  at  any  point  to  the  graph  of  each 
of  the  following  equations.  Find  the  rate  of  change  of  the  slope  m  with 
respect  to  z.  Symbolize,  the  rate  of  change  of  m  with  respect  to  a;  by  F 
and  plot  the  graphs  of  y,  m  and  F  on  the  same  axes.  Find  the  ordinate  of 
the  point  on  the  graph  of  y,  and  also  of  the  point  on  the  graph  of  F,  which 
has  the  same  abscissa  as  the  minimum  point  of  m.  Where  is  the  line 
tangent  to  the  graph  of  y  horizontal? 

(a)  2/  =  x3  -  6x2  +  9x  -  2.  (b)  2/  =  x'  -  3x2  -  9x  +  4. 

—      (c)  2/  =  4x3  -  15x2  ^  i2x.  (d)  2/  =  x'  -  12x  +  5. 

m  Note.  The  point  on  a  graph  where  m  ceases  to  decrease 
and  begins  to  increase  (or  vice-versa)  is  called  a  'point  of  in- 
flection.  The  graph  is  concave  downward  on  one  side  of  this 
point  and  concave  upward  on  the  other  side.  For  this  value  of 
X  the  value  of  F  is  zero. 

Hence,  to  find  the  abscissas  of  the  points  of  inflection  on  the  graph 
of  a  function^  equate  F  to  zero  and  solve  for  x.  Substitute  these 
values  of  x  in  the  equation  y  =  f(x)  to  find  the  ordinates, 

6.  Find  the  coordinates  of  the  point  of  inflection  of  the  graph  of  each  of 
the  following  equations.  Translate  the  axes  to  this  point  as  a  new  origin. 
Show  that  the  graph  is  symmetric  with  respect  to  the  point  of  inflection, 
and  plot  the  graph  of  the  equation  on  the  new  axes. 

(a)  2/  =  a;'  -  3x2  -  6x  +  6,  (b)  2/  =  x3  -  6x2  +  i2x  -  11. 

R  (c)  2/  =  a:^  +  6x2  +  i2x  +  8.  (d)  2/  =  -  a:^  +  3x2  +  X  -  6. 

'  7.  Find  m  and  F  for  the  function  ax^  +  6x2  +  ex  +  d.  jf  ^j^g  g^xes  are 
'  translated  to  the  point  of  inflection  of  the  graph  of  this  function,  show  that 
the  equation  of  the  graph  referred  to  the  new  origin  is  y'  =  ox'^  +  m'x', 
where  m'  is  the  slope  of  the  line  tangent  to  the  graph  of  y  at  the  point  of 
inflection,  and  that  the  graph  is  symmetrical  with  respect  to  the  point  of 
inflection. 

8.  Given  s  =  t^  -  it,  find  (a)  the  average  velocity  from  ^  =  2  to  <  =  4, 
(b)  the  average  of  the  velocities  at  «  =  2,  and  t  =  4,  (c)  the  velocity  at 
t  =  3,  (d)  the  value  of  t  at  which  the  velocity  is  equal  to  the  average  velocity 
for  the  interval  t  =  2  to  t  =  4. 

9.  Find  the  points  of  inflection  of  the  following  functions  and  plot  their 
graphs 

(a)  x4  -  8x»  +  3x2.  (b)  3.6  _  ioa.2  _  g,  (c)  x^  -  4x». 


140  ELEMENTARY  FUNCTIONS 

10.  For  what  positive  integral  values  of  n  does  x"  have  a  point  of  in- 
flection? 

61.  Solution  of  Equations.  Rational  Roots.  Any  algebraic 
equation  in  one  variable  may  be  reduced  to  the  form 

f{x)  =  0,  ^  (1) 

where  f{x)  is  a  polynomial,  by  the  elementary  rules  for  sim- 
plifying an  equation  involving  fractions  and  radicals.  The 
real  roots  of  the  equation  (or  zeros  of  the  function)  are  repre- 
sented graphically  by  the  intercepts  on  the  a;-axis  of  the  graph 
of /(a:).     (See  page  23.) 

If  any  integers  are  roots  of  (1),  they  will  appear  when  the 
table  of  values  for  the  graph  is  being  constructed.  Thus  the 
integral  roots  of  the  equation 

^3  _  3^2  _  lOa;  +  24  =  0 

obtained  by  equating  to  zero  the  polynomial  in  the  example  of 
Section  49  are  2,  4,  and  —  3,  since  the  table  of  values  shows  that 

/(2)=0,        /(4)=0,        and        /(- 3)  =  0. 

That  these  are  all  the  roots  follows  from 

Theorem  1.  An  equation  of  degree  n  has  n  roots,  of  which 
some  may  he  imaginary  and  some  equal  to  each  other. 

This  theorem  is  assumed  without  proof. 

Returning  to  equation  (1),  any  real  roots  other  than  integers 
are  either  fractions  or  irrational  numbers.  In  this  section  we 
shall  see  how  to  find  the  fractional  roots  of  an  equation,  which 
with  the  integral  roots,  constitute  the  rational  roots. 

Theorem  2.  If  x  =  a  is  a  root  of  an  equation  of  degree  n, 
f(x)  =  0,  thenf{x)  =  (x  -  a)q{x),  where  q{x)  is  of  degree  n  -  1. 

By  the  definition  of  a  root  of  an  equation  f(a)  =■  0.  Then 
by  the  remainder  theorem,  if  f(x)  is  divided  by  x  -  a,  the  re- 
mainder is  zero,  so  that  f(x)  is  exactly  divisible  by  x  -  a. 
Since  the  degree  of  the  divisor  is  1,  the  quotient  q(x)  will  be  a 
polynomial  of  degree  n  -  1.  Since  the  dividend  is  equal  to 
the  product  of  the  divisor  and  quotient,  we  have  the  identity 
fix)  -  (X  -  a)q(x),  (2) 


ALGEBRAIC   FUNCTIONS 


141 


Corollary  1.  Any  root  of  the  given  eqvMion  f{x)  =  0,  except 
X  =  a,  is  also  a  root  of  the  equation  q{x)  =  0. 

Let  X  =  h  be.  any  root  of  f(x)  =  0  different  from  x  =  a; 
then  fib)  =  0.     Substituting  a;  =  6  in  (2) 

0  =  (6  -  a)qih). 

But  if  the  product  of  two  numbers  is  zero,  one  of  the  numbers 
is  zero,  and  since  6  -  a  is  not  zero,  we  have  q(b)  =  0,  and  hence 
a;  =  6  is  a  root  of  the  equation  q{x)  =  0. 

Corollary  2.  Any  root  of  the  equation  q{x)  =  0  is  a  root  of 
ike  equation  f{x)  =  0. 

If  a;  =  c  is  a  root  of  q{x)  =  0,  then  q{c)  =  0.  Setting  x  =  c 
in  (2)  we  have 

/(c)  =  (c  -  a)q(c)  =  (c  -  a)-0  =  0. 

Hence  a;  =  c  is  a  root  of  the  equation 
fix)  =0. 

This  last  theorem  and  its  corollaries 
simplify  the  solution  of  many  equations 
hj  enabUng  us,  as  soon  as  a  rational 
root  is  known,  to  replace  the  given 
iequation  by  one  of  lower  degree. 

Example  1.  Solve  the  equation  f(x)  = 
12x3  -  11x2  _  13a.  +  10  =  0.  The  graph  shows 
that  the  three  roots  are  all  real,  and  the  table 
of  values  shows  that  one  root  is  -  1.  Divid- 
ing fix)  by  X  -  (-  1)  =  X  +  1,  we  have  from 
the  table  the  quotient  q{x)  =  12x2  _  23^  +  10. 


u 

y 

.0 

,» 

/ 

. 

. 

/ 

\. 

J/ 

i  1 

0 

^ 

X 

, 

/ 

/ 

/ 

/ 

30 

Fig.  68. 


0 
1 
2 
1 
S.2 


12  -  11  -  13 
+  1-12 
+  13  +  13 
-23  +  10 
-35  +  57 


Kx) 


+  10 
-  2 
+  36 
0 
-104 


By  Corollary  2,  the  roots  of  the  equation 

q{x)  =  12x2  -  23x  +  10  =  0 

are  also  roots  of  /(x)  =  0. 
Factoring  q{x),  we  have 

(3x  -  2)  (4x  -  5)  =  0. 

Hence   3x  -  2  =  0  and  4x  -  5  =  0,  and 


the  roots  are  therefore 

X  -  f  and  X  =  |. 
The  roots  of  the  given  equation  are  therefore  x  -  ~  1,  f ,  and  |. 


142 


ELEMENTARY  FUNCTIONS 


We  note  that  the  denominators  of  the  fractional  roots  in 
Example  1  are  both  factors  of  the  coefficient  of  the  term  of 
highest  degree,  and  that  the  nmnerators  are  factors  of  the  con- 
stant term.  This  conclusion  is  a  special  instance  of  the  fol- 
lowing theorem  which  we  assmne  without  proof. 

Theorem  3.  If  a/h  is  a  root  of  an  equation  with  integral 
coefficients,  then  h  is  a  factor  of  the  coefficient  of  the  highest  power 
of  X  and  a  is  a  factor  of  the  constant  term  (or  of  the  coefficient  of 
the  lowest  power  of  x  if  the  constant  term  is  zero). 

Corollary.  If  the  coeffixnent  of  the  highest  powers  of  x  is  unity 
and  the  other  coefficients  are  integers j  then  the  only  rational  roots 
are  integers. 

Fractional  roots  may  be  determined  by  the  use  of  this 
theorem,  synthetic  division  and  the  remainder  theorem  as  in 

Example  2.     Solve  the  equation  3a;*  -  5x^  -  2^^  +  16x  +  40  =  0. 
The  table  of  values  and  the  graph  are  given  below. 


X 

fix) 

0 

3  _    5-24  +  16 

+    40 

1 

-    2-26-10 

+   30 

2 

+    1-22-28 

-    16 

3 

+    4-12-20 

-    20 

4 

+    7+    4  +  32 

+  168 

-1 

-    8-16  +  32 

+     8 

-2 

-  11  -    2  +  20 

0 

-3 

-14  +  18-38 

+  154 

u 

m 

150 

1 

100 

— 

[^ 

^ 

V 

/. 

-- 

1 

0 

\ 

[  y 

X 

^ 

Fig.  69. 
One  root  is 


2.    Divid- 


The  graph  shows  that  there  are  four  real  roots. 
ing  fix)  by  a;  +  2,  we  have  from  the  table  the  quotient 

qix)  =3x3 -11x2 -2a; +  20. 
By  Corollary  2,  the  roots  of  the  equation  qix)  =0  are  the  other  roots  of 
fix)  =  0.  One  root  Hes  between  1  and  2.  A  fraction  between  1  and  2 
whose  numerator  is  a  factor  of  20  and  whose  denominator  is  a  factor  of 
3  is  |.  Dividing  the  polynomial  by  x  -  f,  it  is  I  "  3  -  11  -  2  +  20 
seen  that  x  =  f  is  a  root  and  that  the  quotient  is     f  i  -    6  -  12       0 

3x2  _  6x  -  12.     Equating  this  function  to  zero,  dividing  by  3,  and  solving 
by  the  quadratic  formula  we  have 

_      2 


V4  +  16 


=  1  =fcV5  =  1  =*=  2.236 


Hence  the  roots  of  the  given  equation  are  x 


=  3.236  or  - 
2, 1,  3.236, 


1.236. 
- 1.236. 


ALGEBRAIC   FUNCTIONS 


143 


Example  3. 


Solve  the  equation 

Qx^  -  16x4  +  ii^z  ^  37a;2  _  116a;  +  60  =  0. 


(3) 


X 

m 

0 
1 
2 
3 
-  1 
-2 

6  -  16  +  11  +  37  -  116 
-10+    1  +  38-    78 
_    4  +    3  +  43  -    30 

+    2  +  17  +  88  +  148 
-22  +  33+    4-120 
-  28  +  67  -  97  +    78 

+    60 

-  16 

0 
+  504 
+  180 

-  96 

f 

6  -    4  +    3  +  43 
+  0  +    3  +  45 

-30 
0 

_    3 

2  +    0  +    1  +  15 
-    3  +  li 

y 

h\ 

160 

1  m 

j 

UP 

\ 

i 

' 

\ 

/ 

1 

\ 

^ 

-• 

?    -2 

-      0 

w 

X 

M 

Fig.  70. 


An  examination  of  the  table  of  values  of  the  function  j{x)  shows  that 
a;  =  2  is  a  root,  and  that  the  quotient  obtained  by  dividing /(x)  by  a;  —  2  is 

q{x)  =  6x*  -  4d;3  +  3x2  +  433.  _  30. 

By  Corollary  2  all  the  roots  of  q{x)  =0  are  identical  with  the  remaining 
)ts  of  the  given  equation.  The  graph  shows  that  one  of  these  is  between 
and  1.  If  this  root  is  rational,  it  must  be  one  of  the  fractions  \,  \,  f 
i  (Theorem  3).  From  the  graph  it  appears  that  the  most  probable  of  these 
values  is  f  and  we  therefore  test  it  first.  As  shown  in  the  table,  if  q{x)  is 
divided  by  x  -  f  the  remainder  is  zero,  and  therefore  x  =  f  is  a  root  of 
f(x)  =  0  and  hence  of  (3). 
Applying  Corollary  1  to  the  equation  q{x)  =  0,  it  is  seen  that  the  re- 
laining  roots  of  this  equation,  and  hence  of  the  original  equation,  are 
Identical  with  the  roots  of 

6x3  4-  3a;  +  45  =  0, 
)r  the  simpler  form  2x'  +    x  + 15  =  0.  (4) 

The  graph  shows  that  there  must  be  a  real  root  of  (3),  and  hence  of  (4), 
between  —  1  and  -  2.  Theorem  1  shows  that  the  only  possibility  of  a 
rational  root  between  —  1  and  —  2  is  —  f.  Dividing  2x'  +  x  +  15  by  x  +  f 
:Kleads  to  a  remainder  that  is  not  zero  and  hence  x  =  -  |  is  not  a  root. 

The  division  of  the  left  member  of  (4)  by  x  +  |  in  the  table  stopped  with 
e  appearance  of  a  fraction  among  the  partial  remainders.     This  is  suf- 
ficient to  show  that  the  final  partial  remainder  cannot  be  zero.     For  if  a 
partial  remainder  is  prime  to  the  denominator  of  a  fractional  divisor  in 
lowest  terms,  it  will  be  prime  to  the  powers  of  that  denominator  which 
ppear  in  the  further  multiplication  by  the  divisor.     Hence  the  remaining 


I 


144  ELEMENTARY  FUNCTIONS 

partial  products  must  be  fractions  and  the  last  integral  coefl&cient  cannot 
be  canceled  by  a  fraction. 

The  rational  roots  of  (3)  are  therefore  x  -  2  and  |,  and  the  irrational 
root  is,  from  the  graph,  nearly  -  1.8.  The  other  two  roots  (Theorem  1) 
are  imaginary. 


EXERCISES 

1.  Find  all  the  roots  of  the  equations  below,  in  each  case  constructing 
the  graph  of  the  polynomial  on  the  left. 

(a)     :j?  -     a;2  -  7x  +   3  =  0.  (b)    a;'  +  5x2  +     a;  _  12  =  o. 

(c)     x^  -    4a:3  +  &c  -    3  =  0.  (d)     7^  -  2x^  ■{- 2x^  -    x  -  6  =  0. 

(e)  6x3  _  Yj^i  +  7a;  _    5^0.  (f)   Zx^  +  4x2  -  21x  +  10  =  0. 

(g)  6x'  +  20x2  +    X  -  20  =  0.  (h)  4x4  +  4x3  -  5x2  -  9x  -  9  =  0. 

(i)  6x4  _    8x3  -  5x2  -  4x  -  4  =  0.  (j)   q^  +  23x3  -  7x2  _  n^;  ^  4  ^  q 

2.  Find  all  the  rational  roots  of  the  equations  below,  in  each  case  con- 
structing the  graph  of  the  polynomial  on  the  left,  and  estimating  the  ir- 
rational roots,  if  any,  from  the  graph. 

(a)  2x*  -  x3  +  4x2  +  24x  -  13  =  0. 

(b)  3x4  +  2x3  +  3a;2  _  iQx  -  14  =  0.  | 

(c)  5x5  _  7a;4  _  a;3  _  12x2  +  ^j  +  6  =  0.  J 

(d)  2x5  +  5x4  -  6a;3  _  n^ja  _  3x  +  9  =  0. 

(e)  6x3  _  i7a;2  +  7x  -  5  =  0. 

(f)  3x4  ^  10x3  +  7x2  -  3x  -  7  =  0.  Using  synthetic  division,  find  the 
irrational  root  to  the  nearest  tenth  of  a  unit. 

3.  Show  that  the  graph  of  a  polynomial  of  degree  n  cannot  have  more 
than  n  -  1  maximum  and  minimum  points,  and  that  it  cannot  have  more 
than  n  —  2  points  of  inflection. 

52.  Translation  of  the  y-Axis.  For  the  purpose  of  determin- 
ing approximately  an  irrational  root  of  an  equation  f{x)  =  0, 
where  f{x)  is  a  polynomial,  a  method  is  used  which  reduces 
the  roots  of  an  equation  by  a  constant  amount  c. 

A  translation  of  the  2/-axis  along  the  x-axis  in  the  positive 
direction  diminishes  the  intercepts  on  the  a;-axis  of  the  graph 
of  f{x),  and  hence  such  a  translation  represents  graphically 
a  diminution  of  the  roots  of  an  equation. 

We  shall  consider  the  method  first  from  a  graphical  point 
of  view. 


ALGEBRAIC   FUNCTIONS  145 

Example.    Construct  the  graph  of 

2/  =  a;3  -  9x2  +  23a;  _  15  (1) 

and  find  the  function  whose  graph  is  the  same  curve  referred  to  a  new  axis 
2  units  to  the  right  of  the  old  one. 

The  graph  is  constructed  by  the  method  of  Section  49. 

The  equations  for  translating  the  axes  are  (Sec- 

tion  31,  page  89) 

a:  =  x'  +  2, 

y  =  y'- 

Substituting  in  (1)  we  have 

y'  =  (x'  +  2)3  -  9(x'  +  2)2  4-  23(x'  +  2)  -  15  (2) 

-  x'3  +  6x'2  +  12x' +    8 

-9x'2-36x'  -36 

+  23x'  +  46 

or  -15 

y'  =  x'3  -  3x'2  -  x'  +  3,  (3) 

which  is  the  required  polynomial. 

If  /(x)  represents  the  function  in  (1),  then 
f{x'  +  2)  will  represent  the  function  in  (2).  The 
intercepts  on  the  x-axis  of  the  graph  referred  to 
the  new  origin  are  2  units  less  than  the  intercepts 
referred  to  the  old  origin.  Hence  the  roots  of  the 
equation  f{x'  +  2)  =  0,  or 

x'3  -  3x'2  -  X  +  3  =  0 

are  2  units  less  than  the  roots  of  the  equation /(x)  =  0  or 

x3  -  9x2 +  23x- 15  =  0. 

We  shall  now  derive  a  method  of  obtaining  the  coefficients  of  (3)  which  is 
simpler  in  general  than  the  preceding  method. 


V 

u' 

16 

\ 

i 

5 

/ 

1 

0 

( 

■1 

V/ 

S    6  xl 

. 

\ 

1/ 

■S 

-IS 

Fig.  71. 


Let 


fix)*  =  OoX**  +  oix^-i  + 


+  an_2X2  +  aw_iX  +  On. 


(4) 


Substituting  x  =  x'  +  c  we  have 

/(x'  +  c)  =  ao(x'  +  c)**  +  ai(x'  +  c)"-*  +  .   .   . 

+  a„_2(x'  +  c)2  +  a„_i(x'  +  c)-\-an  (6) 

Expanding  the  terms  on  the  right  by  the  binomial  theorem,  collecting  like 
powers  of  x',  and  representing  the  coefficients  of  the  powers  of  x'  by 

&o,  hi,  hi,  .  .  .  hn, 

\we  have     f{x'  +  c)  =  feox'"  +  hx'""-^  +  •    •   •  +  6,^2x'2  +  hn-ix'  +  &«.       (6) 

For  any  value  of  x  and  x'  connected  by  the  relation  x  =  x'  +  c, 

or  X  -  c  =  x',  (7) 


146  ELEMENTARY  FUNCTIONS 

we  have  f(x)  =fix'  +  c).  (8) 

Dividing  (8)  by  (7)  -^  =  ^-^^^^'  (9) 

X   —  C  X 

If  fix'  +  c)  =  box'"  +  bia;'"-!  +  .  •  .  +  6n-2x'2  +  6n_ix'  +  6„  be  divided  by 
x',  as  indicated  on  the  right  of  (9),  the  remainder  is  6„  and  the  quotient  is 

qi{x')  =  6oa:''»-i  +  bix'''-^  +  .    .    .  +  bn-2x'  +  bn-i. 
If  the  quotient  qi{x')  is  divided  by  x',  the  remainder  is  ?)r_i  and  the  quotient 
is  q2{x')  =  box'"*-^  +  bix'""-^  +  •   •   •  +  fc«-2. 

Continuing  this  process  it  is  seen  that  the  successive  remainders  are  the 
coefl&cients  of  (6)  in  the  order  bn,  6n-i,  6n-2,  .  .  .  5o.  But  if  the  indicated 
divisions  be  performed  on  both  sides  of  (9),  if  the  respective  quotients  thus 
obtained  be  divided  by  x  -  c  and  x',  and  so  on,  the  successive  remainders 
obtained  on  the  left  side  of  (9)  will  equal  those  obtained  on  the  right, 
namely,  bn,  6n-i,  bn-2,  .  .  .  bo.    Hence  we  have  the 

Theorem.  The  coefficients  of  f{x'  +  c)  may  he  obtained  by 
dividing  J{x)  by  x  —  c,  the  quotient  by  x  —  c,  etc. 

1-9  +23-15  |_2  The  computation  of  the  coefficients 
+  2—14+18  in  the  example  may  be  effected  by 

means  of  this  theorem  and  synthetic 
division  in  compact  form  as  indicated. 
The  successive  remainders  in  the  divi- 
sions, marked  with   an  asterisk,  are 

1*  -  3*  the  coefficients  of  the  function  in  (3). 

EXERCISES 

1.  Construct  the  graph  of  each  of  the  functions  below,  and  find  the 
function  having  the  same  graph  referred  to  a  new  2/-axis  c  units  to  the  right. 
Verify  the  result  by  direct  substitution  of  a;'  +  c  for  x. 

(a)  x'  -  3x2  +  a:  -  2,  c  =  3.  (c)  x*  +  Sx^  -  7x^  +  ISx  -  5,  c  =  2. 

(b)  2x3  _  a;2  +  3,  c  =  1.  (d)  3x^  -  Sa^  -  9x  -  12,  c  =  1. 

2.  Show  that  the  second  term  of  the  function  y  =  x^  -  Qx^  +  7x  +  4i  will 
be  removed  if  the  y-axis  is  translated  2  units  to  the  right. 

Deduce  a  rule  for  removing  the  second  term  of  ax^  +  bx^  +  cx  +  d. 

3.  Determine  a  translation  of  the  2/-axis  so  that  the  graph  ofx^  +  Sx^-4: 
in  the  old  system  will  be  the  graph  of  x'  -  3x  +  2  in  the  new  system.  Plot 
the  graphs  of  x'  and  3x  -  2  on  the  same  axes  and  from  them  determine 
approximately  the  roots  of  the  equation  x'  -  3x  +  2  =  0,  and  hence  of  the 
equation  x'  +  Sx^  -  4  «•  0. 


1-7 

+  2 

+ 

9 
10 

+ 

3* 

1  -5 

+  2 

— 

1* 

ALGEBRAIC  FUNCTIONS 


147 


4.  Plot  the  graph  of  x^  -  2x3  -  Sx^  +  4x  +  2  ^^j^^j  determine  the  co- 
ordinates of  the  maximum  and  minimum  points  and  of  the  points  of  in- 
flection. Find  the  function  which  has  the  same  graph  referred  to  a  ?/-axis 
1  unit  to  the  left. 

53.  Homer's  Method  of  Solution  of  Equations.  This 
method  enables  us  to  compute  irrational  roots  as  acciu'ately  as 
may  be  desired.  j 

Example.  Find,  correct  to  two  decimal  places,  the  real  root  of  the 
equation 

fix)  =  x'  +  X  -  47  =  0.  (1) 

Plotting  the  graph  of /(x)  we  get  the  curve  in  the  figure,  which  shows  that 
there  is  a  real  root  between  3  and  4.    As  the  coeflScients  of  /(x)  are  in- 
tegers,  and  that  of  x^  is  unity,  this  root  is  not  frac- 
tional, and  it  therefore  must  be  irrational. 
Now  move  the  y-sods  3  units  to  the  right. 

1  +  0+1-47  |3 

3+    9+30 
3  +  10-17 

3  +  18 
6  +  28 
3 


The  new  equation,  omitting  the  primes  on  the  x's,  is 

fix  +3)  =  x'  +  9x2  +  28x  -  17  =  0.  (2)      

No  confusion  should  arise  from  omitting  the  primes  Fig.  72. 

if  it  be  remembered  that  the  graph  of  the  poljniomial 

in  (2)  is  the  curve  in  the  figm-e  referred  to  the  axes  O'X  and  O'Y'. 

Equation  (2)  has  a  root  between  0  and  1,  which  from  the  graph  appears 
to  be  about  0.5.     Dividing  (2)  by  x  -  0.5,  we  have 

1  +  9     +28        -17  [0.5 

0.5+   4.75  +  16.375 
9.5  +  32.75  -    0.625 

As  the  remainder  is  negative,  the  graph  lies  below  the  x-axis  at  x  =  0.5 
and  hence,  from  the  figm-e,  the  root  is  larger  than  0.5.     Try  x  =  0.6. 

1  +  9      +28        -17  10.6 

+  0.6+    5.76  +  20.256 
+  9.6  +  33.76+    3.256 

The  remainder  is  now  positive  and  hence  the  graph  lies  above  the  x-axis 
at  X  =  0.6.     Therefore  the  root  of  (2)  lies  between  0.5  and  0.6. 


" 

y 

V 

f 

1 

-. 

o 

-e 

'^'1 

a'  X 

1 

( 

i 

- 

-30 
•36 

/ 

/ 

A 

> 

V 

" 

■^ 

148  ELEMENTARY  FUNCTIONS 

Move  the  ^-axis  0.5  to  the  right. 

1  +  9+28       -17  10.5 

0.5+  4.75+  16.375 
9.5  +  32.75  -  0.625 
0.5+    5 


10     +37.75 
0.5 
10.5 
The  new  equation  is 

a^  +  10. 5x2  ^  37  75^.  _  0  625  =  0.  (3) 

The  graph  of  the  polynomial  on  the  left  is  the  same  curve  referred  to  the 
axes  0"X  and  0"Y",  and  it  shows  that  equation  (3)  has  a  root  between 
0  and  0.1.  As  the  square  and  cube  of  a  number  less  than  0.1  are  very  small, 
an  approximate  value  of  the  root  may  be  obtained  by  neglecting  x'  and  x^ 
in  (3)  and  solving  the  resulting  linear  equation 

37.75X- 0.625  =  0. 
This  gives  x  =  „_'         =  0.02  approximately.     Substitute  0.02  in  the 
left  member  of  (3)  by  dividing  by  a;  -  0.02. 

1  +  10.5    +37.75      -0.625  [0.02 

+    0.02+      .2104  +  0.759208 
+  10. 52  +  37. 9604  +  0. 134208 

As  the  remainder  is  positive,  the  graph  shows  that  the  root  is  less  than 
0.02.     Try  0.01. 

1  +  10.5    +37.75      -0.625  |  0.01 

+    0.01+    0.1051  +  0.378551 
+  10. 51  +  37. 8551  -  0. 246449 

This  remainder  is  negative,  and  hence  the  graph  shows  that  the  root  is 
greater  than  0.01.     Hence  the  root  of  (3)  hes  between  0.01  and  0.02. 

Then  the  root  of  equation  (2)  lies  between  0.51  and  0.52,  and  that  of 
(1)  between  3.51  and  3.52.  Hence  the  real  root  of  (1),  correct  to  two  deci- 
mal places,  is  a;  =  3.51. 

Which  is  the  closer  approximation  to  the  root,  3.51  or  3.52?     Why? 

EXERCISES 

Find  all  the  real  roots  of  the  equations  following,  obtaining  irrational 
roots  to  two  decimal  places. 

1.  a;'  +  2x  -  17  =  0.  6.  x'  -  4^2  +  3x  +  3  =  0. 

2.  x'  +  3x  -  31  =  0.  6.  X*  +  x«  -  15x  +  2  =  0. 

3.  2x'  +  X  -  37  =  0.  7.  x«  -  5x«  +  2x2  +  1  =  0. 

4.  x»  -  6x2  +  8x  +  1  -  0. 


ALGEBRAIC   FUNCTIONS 


149 


Note.  To  find  negative  roots  by  Horner's  method,  replace 
X  in  the  equation  by  -  x.  The  graph  of  /( -  x)  is  symmetrical 
to  that  oi  f{x)  with  respect  to  the  y-sods,  and  the  roots  of  the 
equation  f{—x)  =0  will  be  equal  numerically  to  those  of 
f(x)  =  0,  but  have  opposite  signs.  Hence  the  negative  roots 
of  an  equation  f(x)  =  0  may  be  found  by  finding  the  positive 
roots  of  f(-x)=  0,  and  changing  their  signs. 


8.  a;3  ^.  2x  -f  23  =  0. 

9.  a;'  +  x2  +  7  =  0. 

10.  x3  -  x2  -  6x  +  1  =  0. 


11.  x4  _  32.3  _  4^.2  +  i2x  -  10  =  0. 

12.  x^  -  2a:'  -  7  =  0. 


Note.  If  an  equation  has  both  rational  and  irrational  roots, 
it  is  advisable  to  find  the  rational  roots  first.  Suppose  they 
are  a,  fi,  7,  etc.  Then  divide  the  equation  by  x  —  a,  the  re- 
sulting equation  by  x  —  j3,  the  new  equation  by  x  —  7,  etc., 
thus  obtaining  a  simpler  equation  whose  irrational  roots  are 
the  same  as  those  of  the  given  equation,  and  then  solve  this 
simpler  equation  by  Horner's  method. 

13.  a:*  -  3a;3  +  a:2  +  1  =  0. 

14.  x^  +  2x3  +  3a;2  _  43a.  _  93  =  0. 

15.  3x5  -  x*  +  4a;3  _  i6a.2  _  333.  +  13  =  Q. 

16.  6x*  -  31x3  ^  40x2  +  2x  -  5  =  0. 

17.  x^  -  3x3  -  4x2  +  14x  -  6  =  q. 

18.  6x5  ^  17^  +  4x'  -  39x2  -  297x  +  210  =  0. 

19.  Find  the  fifth  root  of  279. 

20.  A  cast  iron  rectangular  girder  (breadth  =  \  depth)  rests  upon 
supports  12  feet  apart  and  carries  a  weight  of  2000  pounds  at  the  center. 
In  order  that  the  intensity  of  the  stress  may  nowhere  exceed  4,000  pounds 
per  square  inch,  it  is  determined  that  the  depth  d  of  the  girder  in  inches 
must  satisfy  the  equation  SOd'  —  81^2  —  17,280  =  0.  Find  d  and  the  cross- 
sectional  area. 

21.  The  depth  of  flotation  of  a  buoy  in  the  form  of  a  sphere  is  given  by 
the  equation  x^  -  3rx2  +  4r3s  =  0,  where  r  is  the  radius  and  s  is  the  specific 
gravity  of  the  material.  What  is  the  depth  for  such  a  buoy  whose  radius 
is  1  foot  and  specific  gravity  is  0.786? 

22.  The  cross  section  of  the  retaining  wall  of  a  reservoir  is  designed  as 
indicated  in  Fig.  73.  The  allowable  height  x  of  the  upper  portion  is 
given  by  the  equation  x'  4-  32x2  _  qq^  -  88  =  0,  where  x  is  expressed  in 
terms  of  a  unit  of  10  feet.     Find  the  allowable  height  to  three  significant 

ires. 


150 


ELEMENTARY  FUNCTIONS 


23.  The  allowable  height  of  the  lower  portion  of  the  wall  in  Exercise  22 
is  given  by  the  equation  y^  -  2.7 y^  +  18.4i/2  _  I23i/  -  1.46  =  0,  where  y  is 
in  terms  of  a  unit  of  10  feet.  Find  the  height  of  the  lower  portion  to 
three  significant  figures. 

24.  In  Exercise  22  the  slope  of  the  water  front 
BC  to  the  vertical  is  jV,  of  J^E  is  -^^,  of  AE  is  xV^. 
The  width  of  the  top  is  6  feet.  What  must  be  the 
width  of  the  lower  portion? 

25.  The  load  P,  concentrated  at  the  center, 
which  a  homogeneous  elliptical  plate  can  support 
is  given  by  the  formula 

3  +  2w2  +  3m4 


Fig.  73. 


8m 


8  +  4m2  +  3m* 


h'^R, 


where  h  is  the  thickness  of  the  plate  =  ^q  \tl.,  R  is  the  maximum  safe  unit 
stress  for  the  material  =  16,000  pounds  per  square  inch,  P  =  6007r,  and 
m  is  the  ratio  of  the  breadth  to  the  length  of  the  elhpse.  Find  the  value 
of  m. 

26.  The  maximum  stresses  on  a  parabolic  arch  of  a  bridge  are  given  by 
the  roots  of  the  equation  2r^  -  5r*  +  Or'^  +  8r  +  2  =  0,  where  r  is  the  ratio 
of  the  length  of  the  arch  occupied  by  a  moving  load  such  as  a  train.    Find  r. 

54.   Graph  of  the  Function  f{ax).     Before  proceeding  to  the 
summary  in  the  next  section,  we  shall  see  how  the  graph  of 
f{ax)  may  be  found  from  that  of  fix).    This  will  complete  the 
study  we  shall  make  of  pairs  of  re- 
lated functions  and  their  graphs.    Con- 
sider the 

Example.  If  f{x)  =x^  -  6x,  then  j{2x) 
=  (2x)2  -  6(2x).  Show  that  the  graph  of 
/(2a;)  may  be  obtained  by  bisecting  the  ab- 
scissas of  points  on  the  graph  off(x). 

If  we  substitute  4  for  x  in  the  first  func- 
tion, and  half  of  4,  namely  2,  in  the  second, 
the  results  are  both  equal  to  -  8.  Hence  the 
point  (4,  -8)  lies  on  the  first  graph  and 
(2,  -  8)  on  the  second,  and  the  abscissa  of 
the  latter  point  is  half  that  of  the  former. 

If  we  substitute  any  value  for  x  in  the  first 
function,  and  half  that  value  in  the  second, 
the  results  will  be  the  same,  namely  x^  -  6x. 
Hence  if  (xi,  yi)  is  a  point  on  the  first  graph 
the  point  (a;i/2,  t/i)  will  be  on  the  second.  Fig.  74. 


T 

1  f 

f<!' 

)      f(<^> 

' 

- 

/ 

/ 

\s 

/ 

WT 

/ 
' 

1 

oi 

3 

« 

\             ' 

'X 

u 

*1\ 

/ 

1 

/ 

I  n 

/ 

/ 

\ 

/ 

/ 

J[  \ 

\ 

/ 

/ 

' 

\ 

/ 

\) 

\/ 

/ 

ALGEBRAIC   FUNCTIONS  151 

As  the  latter  point  may  be  obtained  by  bisecting  the  abscissa  of  the  former, 
it  follows  that  the  graph  of /(2x)  may  be  obtained  by  bisecting  the  abscissas 
of  several  points  on  the  graph  of /(x)  and  drawing  a  smooth  curve  through 
them. 

The  reasoning  employed  in  this  example  may  be  used  to 
prove  the 

Theorem.  The  graph  of  f(ax)  may  be  obtained  by  dividing  by 
a  the  abscissas  of  points  on  the  graph  of  f{x).  Corresponding 
paints  on  the  two  graphs  lie  on  the  same  or  opposite  sides  of  the 
.,  y-Qjxis  according  as  a  is  positive  or  negative. 

This  theorem  may  also  be  established  as  follows:  Let 
y  =  f(x)  and  suppose  that  the  result  of  solving  this  equation 
for  X  is  X  =  (j)(y).  The  graphs  of  these  two  equations  are 
identical.  If  we  solve  the  equation  y  =  f{ax)  for  ax  the  re- 
sult must  be  ax  =  </)(2/),  and  the  graphs  of  these  two  equations 

are  identical.    But  the  last  equation  may  be  written  x  =  -(l>{y), 

from  which  it  follows  that  the  abscissas  of  points  on  this 
curve  are  one-ath  of  the  abscissas  of  the  points  on  the  graph 
of  (l>{y). 

This  theorem  will  find  appUcation  in  the  study  of  some  of 
the  transcendental  functions.  It  is  also  applied  in  an  alter- 
native method  of  finding  the  rational  zeros  of  a  polynomial 
in  Exercises  2  to  6  below. 

If  a  has  such  a  value  as  f ,  the  division  of  the  abscissas  by  J 
amounts  to  multiplication  by  3. 

55.  Related  Functions  and  their  Graphs.  We  have  studied 
several  functions  which  may  be  obtained  by  transforming,  or 
changing,  a  given  function,  and  whose  graphs  may  be  obtained 
from  that  of  the  given  function  by  simple  geometric  construc- 
tions or  transformations.  These  are  given  in  the  tables  below. 
In  order  that  the  statements  may  be  concise  and  accurate,  it 
is  assumed  that  the  constants  involved  in  the  table  are  positive. 
The  changes  necessary  if  the  constants  are  negative  should 
cause  no  difficulty. 


152 


ELEMENTARY  FUNCTIONS 


The  graph  of 

may  be  obtained  from  the  graph  of  f{x)  by 

(1) 

fix)  +  k 

moving  it  up  k  units  (page  19,  Exercise  3) 

(2) 

fix  +  h) 

moving  it  to  the  left  h  units  (Theorem  2,  page  92) 

(3) 

am 

multiplying  ordinates  by  a  (Theorem,  page  89) 

(4) 

Kox) 

dividing  abscissas  by  a  (Theorem,  page  151) 

(5) 

1 

by  the  principles  for  the  graphs  of  reciprocal  functions 

Six) 

in  Section  40  (page  117). 

The  graph  of 

(3a) 

-Kx) 

(4a) 

fi-x) 

(4b) 

-f(-x) 

(6) 

inverse  of 

fix) 

is  symmetrical  to  the  graph  of  fix)  with  respect  to 

the  X-axis 
the  i/-axis 
the  origin 

the  bisector  of  the  first  and  third  quadrants  (Theorem, 
page  114). 


It  should  be  noticed  that  (3a)  and  (4a)  are  the  special  cases 
of  (3)  and  (4)  respectively,  obtained  when  a  =  -  1,  and  that 
(46)  may  be  regarded  as  a  combination  of  (3a)  and  (4a). 

Properties  (1)  and  (2)  are  closely  related  to  the  translation 
of  the  axes.     If  we  set  (Theorem  1,  page  89) 

X  =  x'  -hh,  y  =  y'  +  k 
in  y=f(x), 

we  get  y^+k  =  f{x'  +  h). 

The  graphs  of  these  two  functions  are  identical  if  x'  and  y' 
are  plotted  on  axes  h  units  to  the  right  and  k  units  above  the 
old  axes.    But  the  second  equation  may  be  written 

y'  =f{x'  +  h)-k', 

and  hence,  if  x'  and  y'  are  plotted  on  the  old  axes  the  graph  of 
y'  may  be  obtained  by  moving  that  of  y  to  the  left  h  units  and 
k  units  down  (properties  (1)  and  (2)). 

Properties  (3)  or  (4)  may  also  be  interpreted  as  giving  es- 
sentially the  same  distortion  to  a  curve  as  is  obtained  if  un- 
equal units  are  chosen  on  the  coordinate  axes. 

With  these  properties  might  be  associated  the  method  of  ob- 


ALGEBRAIC   FUNCTIONS  153 

taming  the  graph  of  y  =  f{x)  =*=  g{x)  by  first  plotting  the  graphs 
off(x)  and  g(x),  and  then  adding  or  subtracting  the  correspond- 
ing ordinates  (see  Exercise  4  of  the  Miscellaneous  Exercises 
following  Chapter  I). 

56.  Some  Operations  of  Algebra  regarded  as  Properties  of 
Functions.     The  identity  (a  +  by  =  a^  +  2ab  +  ¥ 

is  usually  thought  of  in  elementary  algebra  as  a  rule  for  ob- 
taining the  square  of  a  binomial.  It  may  also  be  regarded  as 
a  property  of  the  function  x^,  that  is,  as  the  means  of  expressing 
the  value  of  the  function  when  x  is  the  sum  of  two  given  num- 
,  bers  in  terms  of  the  squares  and  first  powers  of  the  separate 
nimibers. 

It  will  be  well  to  group  together  a  few  such  relations,  whether 
we  regard  them  as  rules  of  operation  or  as  properties  of  func- 
tions. As  typical  of  such  relations  we  select  the  following 
properties  of  the  function  a;": 

If  fix)  =  x"",  where  n  is  a  positive  integer,  then 

(1)  /(a  +  6)  =  (a  +  6)-  -  a»  +  »a»-ft  +  "    '   '  +  ^  Binomial  theorem. 

(2)  /(a  -  6)  =  (a  -  6)"  =  a"  -  na"  16  +  •   •    •  ±  fe"  j 

(3)  f(ah)       =  (ab)**  =  a«5". 

(4)  f(a/b)     =  (a/5)'»  =  a»/b\ 

(5)  /(aP)_     =  (aP)«  =  a«P. 

(6)  /(  </a)    =  (  \/a)^  =  a'*'^ 

(7)  f(a)  +f{b)  =  a"  +  6"  =  (a  +  6)  (a^-i  -  a""^?)  +  •    •    •  +  6"-^),  if  n  is  odd. 

(8)  /(a)  -fib)  =a^  -b^^ia-b)  (a^-i  +  a^-^b  +  •   •   •  +  b^-^). 

These  relations,  or  special  cases  of  them,  are  used  constantly 
in  transforming  algebraic  expressions,  for  example,  in  simplify- 
ing complex  fractions. 

Relations  analogous  to  some  of  these  will  be  derived  for  each 
of  the  transcendental  functions  to  be  studied  in  later  chapters. 
In  order  to  perceive  the  analogy  clearly,  it  is  desirable  to  as- 
sociate with  these  relations  the  general  notation  in  terms  of 
fix)  given  at  the  beginning  of  each  line. 

Essential  differences  between  various  functions  lie  in  the 
differences  in  analogous  properties. 

A  given  function  may  not  possess  properties  analogous  to 
all  of  the  eight  relations  above.     For  example,  if  f{x)  =  a;^, 


154  ELEMENTARY  FUNCTIONS 

then  /(a)  +  f{b)  =  a^  +  b^,  which  cannot  be  factored  unless  we 
use  imaginary  numbers,  so  that  this  function  has  no  analogue 
to  (7)  in  the  field  of  real  numbers.  Again,  if  f{x)  =  Vx^  then 
f(a  +  6)  =  Va  +  b,  and  there  is  no  other  simple  form  in  which 
this  may  be  expressed,  so  that  Vx  has  no  simple  property  anal- 
ogous to  (1). 

The  connection  between  these  relations  and  the  graphical 
study  of  functions  is  not  as  remote  as  it  might  at  first  seem 
to  be.  For  example,  if  in  (3)  we  set  6  =  -  1,  we  get  /(-  a) 
=  (-  ay  =  (-  l)"(a)"  =  =fc  a"  =  =«=  /(a)  according  as  n  is  even  or 
odd.  This  relation  is  the  one  which  estabUshes  the  symmetry 
of  the  graph  of  x"". 

If  we  replace  a  by  x  in  all  the  relations  above,  it  is  seen  that 
relations  (1)  and  (2)  may  be  associated  with  property  (2)  of 
the  preceding  section,  relations  (3)  and  (4)  with  property  (4), 
and  relations  (7)  and  (8)  with  property  (1).  As  regards  re- 
lations (5)  and  (6),  it  would  be  possible  to  obtain  connections 
between  the  graph  of  f{x)  and  those  of  f{x^)  and  fi\^x),  but 
they  would  be  complicated  and  have  no  important  application. 

EXERCISES 

1.  If  f(x)  =  x^  -  1,  find  /(2x)  and  f(x/2).  Construct  the  graphs  of  the 
three  functions  on  the  same  axes. 

2.  If  f(x)  =  6x2  -  X  -  1,  fin(j  f(x/Q),  and  plot  the  graphs  of  both  func- 
tions on  the  same  axes.  What  relation  will  exist  between  the  intercepts 
of  the  two  graphs  on  the  x-axis?  between  the  roots  of  the  two  equations 
f(x)  =  0  and  /(x/6)  =  0?  Check  your  answer  by  solving  both  equations 
and  comparing  their  roots. 

3.  Iif(x)  =  4x2  _  8x  +  3,  find/(x/2),  and  proceed  as  in  2. 

4.  If  }(x)  =  3x3  _  2x2  +  6x  -  4,  find  /(x/3)  and  9/(x/3).  Plot  the 
graphs  on  the  same  axes,  obtaining  the  graph  of  the  second  function  from 
the  first,  and  that  of  the  third  function  from  the  second.  Find  the  rational 
root  of  9/(x/3)  =  0,  and  from  it  get  the  rational  root  of  /(x)  =  0.  Check 
the  result  by  substitution. 

5.  Replace  x  by  x/a  in  the  equation  ax^  +  bx^  +  ex  +  d  =  0,  and  multiply 
both  sides  of  the  resulting  equation  by  a\  If  o,  b,  c,  d  are  integers,  can 
the  new  equation  have  any  fractional  roots?  If  the  integral  roots  of  the 
new  equation  are  found,  how  can  the  rational  roots  of  the  given  equation 
be  determined  from  them?  Can  a  similar  process  be  used  with  any  equa- 
tion in  the  form  of  a  polynomial  equal  to  zero? 


ALGEBRAIC   FUNCTIONS  155 

6.  Solve  equations  1(e)  to  (j),  page  144,  by  the  method  indicated  in  the 
preceding  exercises. 

7.  Exhibit  in  tabular  form  the  properties  of  x"^^  a;',  1/x,  and  Vx  analogous 
tD  relations  (1)  to  (8),  Section  56. 

MISCELLANEOUS  EXERCISES 

1.  In  the  following  table,  p  represents  the  percentage  of  alcohol  in  a 
p  I  10,  40,  70,  90,  100,  mixture  whose  specific  gravity  is  s.  Find 
s   I  982,  939,  871,  823,  794,     s  as  a  quadratic  function  of  p. 

2.  In  the  table  the  quantity  of  com,  q,  is  assumed  to  be  connected  with 
the  price  p  by  a  relation  of  the  form 

g  I  1,  0.9,  0.8,  0.7  a 

p  I  1,  1.3,  1.8,  2.6  -  ^"  isi-hY 

Plot  the  graph  and  determine  the  values  of  the  constants  a  and  6. 
Hint*  If  two  pairs  of  values  of  p  and  q  are  substituted,  the  resultiiffe  equa- 
tions may  be  solved  for  a  pair  of  values  of  a  and  6. 

3.  In  doing  a  certain  piece  of  work,  a  contractor  paid  a  dollars  per  day 
for  wages,  so  that  the  total  amount  expended  for  wages  in  t  days  was 
TT  =  at.  The  total  amount  invested  in  t  days  increased  according  to  the 
law  /  =  at^l2.  Plot  the  graphs  of  these  equations,  assuming  a  numerical 
value  for  the  wage  rate  a.  Find  the  values  of  W  and  /  if  the  work  is 
finished  in  ^i  days.  What  would  be  the  effect  on  the  wage  rate  if  the  num- 
ber of  workmen  was  increased  so  that  the  work  was  finished  in  <i/2  days? 
;If  the  amount  invested  follows  the  same  law  as  before,  i.e.,  is  haK  the  wage 
rate  times  the  square  of  the  time,  what  would  be  the  value  of  /  if  the  work 

finished  in  ii/2  days? 

4.  It  is  estimated  that  the  quantity  of  work  done  by  a  man  in  an  hour 
varies  directly  as  his  pay  per  hour  and  inversely  as  the  square  root  of  the 
number  of  hours  he  works  per  day.  A  man  can  finish  a  piece  of  work  in 
six  days  when  working  9  hours  a  day  at  $.50  per  hour.  How  many  days 
will  he  take  to  finish  the  same  piece  of  work  when  working  12  hours  a  day 
at  $.55  per  hour? 

5.  An  automobile  manufacturer  estimates  that  if  he  charges  $800  for 
car  he  will  sell  4000  a  month,  and  that  for  each  decrease  of  $100  in  price 

the  sales  will  increase  1500  cars  per  month.  What  price  per  car  will  bring 
the  greatest  gross  returns? 

6.  The  height  of  a  ball  thrown  vertically  upward  with  a  velocity  of 
64  feet  per  second  is  given  by  s  =  -  16^^  ^  54^^  A  boy  in  a  window  16  feet 
above  the  ground  catches  the  ball  as  it  falls.     With  what  velocity  was  it 

oving  when  he  caught  it? 

7.  An  automobile  with  good  brakes  moving  on  a  smooth  dry  pavement 
t  the  rate  of  v  loiles  per  hour  was  stopped  in  s  feet.     Find  the  law  con- 


156^  ELEMENTARY  FUNCTIONS 

10      Ti     90     ^^cting  s  and  v  for  the  corresponding  values 

—  — ^ ^ — ^ — ^'     given  in  the  table.     In  how  many  feet  should 

'       '        '         *        '     the  car  stop  when  moving  30  miles  an  hour? 

8.  The  magnitude  of  the  pressure  on  a  surface  perpendicular  to  the 
direction  of  the  wind  is  proportional  to  the  area  of  the  surface  and  to  the 
square  of  the  velocity  of  the  wind.  If  the  pressure  on  the  side  of  a  build- 
ing 20  feet  wide  and  60  feet  high  is  648  pounds  when  the  velocity  of  the 
wind  is  15  feet  per  second,  what  is  the  pressure  on  a  building  30  feet  by 
80  feet  when  the  velocity  is  45  feet  per  second?  Draw  a  graph  to  show  how 
the  pressure  on  a  particular  building  changes  as  the  velocity  of  the  wind 
changes.  Draw  a  graph  to  show  how  the  pressure  on  various  buildings 
differs  for  the  same  wind. 

9.  The  area  of  the  safety  valve  for  a  boiler  may  be  determined  by 
allowing  1  square  inch  of  valve  to  every  2  square  feet  of  grate  surface. 
Find  the  diameter  of  the  valve  as  a  function  of  the  area  of  the  grate,  and 
plot  the  graph.  What  should  the  diameter  be  for  a  grate  10  feet  by  6 
feet?      • 

Note.  When  any  thin  plane  surface  is  moved  through  the  air  so  that 
the  direction  of  the  motion  makes  a  small  angle  with  the  lower  side,  the 
resultant  pressure  of  the  air  is  very  nearly  normal  to  the  plane  surface. 
The  lifting  efl5ciency  of  the  plane  is  increased  by  curving  it  longitudinally, 
the  concave  surface  being  placed  so  as  to  meet  the  air.  There  is  a  suction 
on  the  upper  surface  of  good  wings  which  amounts  to  more  than  two- 
thirds  of  the  total  lift. 

If  a  plane  surface  is  presented  edgewise  to  the  direction  of  motion, 
there  is  no  upward  pressure,  but  if  a  curved  wing  is  presented  edgewise 
there  is  upward  pressure.  If  the  forward  edge  of  the  wing  is  depressed, 
the  upward  pressure  decreases.  When  it  is  depressed  to  the  point  where 
the  upward  and  downward  pressures  balance,  a  horizontal  chord  of  the 
wing  may  be  drawn  through  the  rear  edg-e.  In  any  other  position  of  the 
wing,  the  angle  which  this  chord  makes  with  the  horizontal  is  called  the 
angle  of  incidence,  6. 

The  equation  connecting  the  normal  pressure  P,  in  pounds,  the  area  S 
of  the  wing  in  square  feet,  the  velocity  v,  in  feet  per  second,  and  the  angle 
of  incidence  6,  in  degrees,  is 

P  =  KSvW, 

where  X  is  a  constant  of  proportionality  which  remains  constant  only  for 
small  values  of  and  variations  in  0. 

If  W  represents  the  weight  of  the  aeroplane  in  pounds,  the  necessary 
condition  for  flight  is  that  the  vertical  component  of  the  pressure  on  the 
plane  is  equal  to  W.  For  small  values  of  d,  the  vertical  pressure  is  very 
nearly  equal  to  the  normal  pressure,  so  that  approximately,  for  hori- 
zontal flight 

W"  KSi/^e.  (1) 


ALGEBRAIC   FUNCTIONS  157 

T  10.  (a)  liW  =  2500  pounds,  S  =  400  square  feet,  and  K  =  0.000087,  plot 
the  graph  of  y  as  a  function  of  6.  In  practice  6  varies  from  about  3°, 
the  lowest  safe  value,  to  12°.  For  what  value  of  6  will  the  aeroplane  have 
the  greatest  horizontal  velocity?  If  additional  power  is  given  by  the 
motor  the  aeroplane  will  not  increase  its  hoiizontal  velocity  but  will  rise, 
and  if  the  motor  is  shut  ofif,  the  aeroplane  begins  to  fall,  but  the  horizontal 
velocity  remains  the  same.     Explain. 

(b)  The  term  loading  is  applied  to  the  quantity  W/S.  If  6  =  5°  and 
K  =  0.000087,  plot  the  graph  of  t;  as  a  function  of  the  loading.  How  will 
V  change  if  S  is  constant  and  W  is  quadrupled?  How  will  v  change  if  W 
is  constant  and  S  is  reduced  by  5? 

(c)  If  W,  S  and  6  have  the  values  used  above,  how  will  v  vary  as  K, 
the  efficiency  coefficient,  changes?  As  the  efficiency  decreases,  will  the 
power  required  increase  or  decrease?  Why  not  make  aeroplanes  with  a 
lifting  coefficient  that  is  inefficient  and  fly  very  fast? 

^11.  The  total  resistancfe  to  the  motion  of  the  aeroplane  is  made  up  of 
two  parts,  the  horizontal  component  of  the  pressure  on  the  planes,  given 
by  Wd,  and  the  resistance  of  the  struts,  body,  etc.,  given  by  csv^,  where  c 
is  a  constant  and  s  is  a  theoretical  surface  which  would  have  the  same  re- 
sistance as  the  various  elements  of  the  aeroplane  except  the  wings.  If 
the  value  of  v  from  (1)  is  substituted,  the  total  thrust,  t,  is 

For  a  given  aeroplane  W,  c,  s,  K,  S  are  constant.  The  single  constant 
/,  called  the  fineness,  is  defined  by  the  equation  1/P  =  cs/KS.  Making 
this  substitution,  we  have 

t-w{e+^g}  (2) 

If  /  =  5,  and  W  =  2500,  plot  t  as  a  function  of  6.  Determine  approxi- 
mately the  most  efficient  value  of  6.  How  does  an  increase  in  S  affect 
the  ratio  S/s,  hence  the  fineness,  and  hence  the  most  efficient  angle? 
How  does  the  thrust  vary  ae  the  fineness  increases?  How  does  the  thrust 
vary  with  respect  to  s?  with  respect  to  the  plane  area,  S?  the  lifting 
efficiency? 
v^J.2.  The  power  required  for  horizontal  flight  at  velocity  p  is 

P-vw(0+^^-  (3) 

Substitute  the  value  of  v  obteined  from  (1)  in  (3)  and  plot  P  as  a  function 
of  6.  As  0  decreases  and  approaches  the  critical  angle  3°,  which  increases 
most  rapidly,  P,  t,  or  »?  What  can  be  said  of  the  power  required  for  high 
speeds  at  small  angles?  How  is  P  affected  by  an  increase  in  Wl  a  de- 
.  crease  in  S7  an  increase  in  K?  an  increase  in  /? 


CHAPTER  IV 


TRIGONOMETRIC  FUNCTIONS  AND  THE  SOLUTION   OF 
TRIANGLES    (ANGLE   MEASUREMENT) 

57.  Introduction.  One  of  the  most  useful  applications  of 
mathematics  has  been  the  systematic  location  and  relocation 
of  points  and  lines  on  the  earth's  surface,  and  the  measure- 
ment of  portions  of  the  surface  —  the  art  of  surveying.  In 
surveying  an  extensive  region,  as  a  state,  two  points  several 
miles  apart,  where  the  intervening  surface  is  level,  are  selected 
for  the  extremities  of  a  so-called  base  line.  The  length  of  this 
base  line  is  measured  with  great  accuracy.  A  number  of 
stations  are  selected  to  serve  as  the  vertices  of  a  network  of 
triangles  stretching  across  the  country,  and  the  angles  sub- 
tended at  each  station  by  every  pair  of  visible  stations  are  care- 
fully measured.  From  these  measurements  the  sides  of  all 
the  triangles  can  be  calculated. 

The  principle  underlying  the  com- 
putations of  the  surveyor  is  that  the 
homologous  ddes  of  similar  triangles  are 
proportional.     Its  use  is  illustrated  in 

Example  1.     To  find  the  width  of  a  river. 

Take  a  base  line  AC  &  short  distance 
from  one  bank,  and  measure  its  length. 
Note  the  point  B  directly  opposite  C,  so  that 
Z.ACB  is  a  right  angle.  From  a  point  B' 
on  AB  draw  B'C  perpendicular  to  AC,  and 
Since  the  right  triangles  ACB  and  AC'B'  are 


measure  AC  and  B'C. 

similar, 

BC 
AC 


B  C  r>^      B  C    .  ^ 

_,  or  BC-3jp,  AC. 


The  width  would  then  be  obtained  by  subtracting  from  BC  the  distance 
CD  from  C  to  the  bank  of  the  river. 

158 


TRIGONOMETRIC  FUNCTIONS 


159 


The  solution  of  the  example  depends  upon  finding  the  ratio 
of  two  sides  of  a  right  triangle  which  has  the  same  acute  angle 
BAC  as  the  given  triangle.  It  would  be  tedious,  and  some- 
times diflScult  or  impossible,  if  every  time  we  wished  to  deter- 
mine an  inaccessible  side  of  a  triangle  we  had  to  construct  a 
similar  right  triangle  whose  sides  could  be  measured  and  their 
ratio  determined.  To  avoid  this  difficulty,  tables  have  been 
constructed  which  give  the  ratios  of  the  sides  of  a  right  triangle 
with  a  given  acute  angle.  Let  us  consider  the  construction  of 
such  a  table. 

Suppose  that  one  acute  angle  of  a  right  triangle  is  30°,  and 
that  the  triangle  be  placed  in  the  position  OMP,  with  the 
vertex  of  this  angle  at  the  origin  of  a  system  of  coordinates, 
the  adjacent  leg  lying  along  the 
positive  part  of  the  x-axis,  and 
the  hypotenuse  falling  in  the  first 
quadrant.  The  triangle  is  placed 
in  this  position  so  that  the  pro- 
cedure will  be  in  accord  with 
fundamental  definitions  to  be 
given  in  Section  59. 

Let  the  coordinates  of  P  be 
X  =  OM  and  y  =  MP,  and  let 
OP  =  r.  We  seek  the  values  of 
the  rations  y/r,  x/r,  and  y/x.  Fig.  76. 

Since  the  angle  at  0  is  30°,  that 
at  P  is  60°.     Hence  AOMP  is  half  an  equilateral  triangle  OPQy 
and  therefore        ^  ^  2y      and      x'  +  y'  =  7^,  (1) 

The  ratios  required  may  be  determined  by  solving  these 
luations  for  any  two  of  the  sides  in  terms  of  the  third.     We 
ilready  have  r  in  terms  of  y.      Substituting  in   the  second 
luation,       a;2  +  2/2  =  4?/2,       whence      x  ^  Vs  y. 
We  may  now  find  the  values  of  the  ratios: 

y  ^y_^l     X  ^  Vsy  ^  Vs     y^     y     ^  _1^  ^  V3      (2) 
r     2y     2'     r        2y         2  '    ^     VSy     Vd       3 


^ 

? 

r/^ 

y 

^^ 

0 

M 

X 

^ 

^^^^ 

^^> 

Q 

160 


ELEMENTARY  FUNCTIONS 


As  all  right  triangles  whose  acute  angles  are  30°  and  60°  are 
similar,  these  results  are  true  for  all  such  triangles. 

Problems  in  geometry  which 
can  be  solved  by  means  of  equa- 
tions (1)  may  be  solved  more 
expeditiously  by  means  of  the 
ratios  (2),  as  in 

Example  2.  Find  the  height  of  a 
flag  pole  MP  if  the  pole  subtends  an 
angle  of  30"  at  a  point  0  100  feet  from 
the  foot  M. 


y 

' 

.f*" 

Ty^ 

y 

>^^ 

• 

0 

X-^IOO 

M 

X 

Fig.  77. 


By  the  method  of  plane  geometry  we  obtain  the  equations 

r  =  2y,     1002  +  y^  =  r\ 

The  solution  is  completed  by  solving  these  equations  for  y. 
Using  the  third  of  the  ratios  (2),  which  contains  the  known  and  required 
sides,  we  have 


2/        V3      ,                  100V3 
^-  =  -5-,  whence  y 5 — 


100x1.732 


100 


=  57.7  feet. 


The    values    of 
and  y/x  for  two 
readily    obtained 
plane  geometry. 

If  Z  MOP  =  45^ 

y  =  X      and 

These  equations 
and  r  in  terms  of 
P 


the   ratios   y/r,  x/r,    y^ 
other  angles  may  be 
by    the    methods    of 

we  have 

rr2  +  2/2  =  r2.        (3) 

may  be  solved  for  y 
X,  and  the  values  of 

the  ratios  determined. 
If  ZMOP  =  60°,  we  have 

r  =  2x      and      y^  +  x^  =r2,     (4) 

which  may  be  solved  for  y  and  r  in 
terms  of  a;,  and  the  ratios  determined. 
The  details  of  the  work  are  left  as 
exercises. 


TRIGONOMETRIC  FUNCTIONS 


161 


e=  Z  atO 

y 

r 

X 

r 

g 

X 

30** 

U.50 

^  =  O.ST 

f-..8 

45** 

f  =  0.71 

f-0.71 

\-uoo 

60° 

f.0.87 

i-0.50 

V3  =  1.73 

The  values  of  the  ratios  for  the  three  angles  already  con- 
sidered are  collected  in  the  table. 
In  computing  the  decimal  fractions,  we  have,  for  example, 
V|^  1.732 
2 


=  0.866. 


The  first  two  decimal  places  are  0.86,  but  0.87  is  a  better 
approximation  to  two  figures. 


162 


ELEMENTARY  FUNCTIONS 


Approximate  values  of  the  ratios  for  any  acute  angle  6  may 
be  found  as  follows:  By  means  of  a  protractor  construct  the 
angle  6  so  that  the  vertex  is  at  the  origin,  one  side  lying  along 
the  X-axis,  and  the  other  in  the  first  quadrant.  From  any 
point  P  on  the  latter  side  drop  the  Une  PM  perpendicular  to 
the  X-axis.  Measure  the  lengths  of  the  sides  of  the  triangle 
OMP  to  find  X,  y,  and  r,  and  then  divide  y  hy  r,  x  by  r,  and  y 
by  X.  Fig.  80  simpHfies  and  systematizes  the  process.  It  con- 
sists of  a  portion  of  a  circle,  whose  center  is  the  origin  and 
whose  radius  is  10,  and  radii  making  angles  of  10°,  20°,  30°, 
etc.,  with  the  x-axis. 

To  find  the  values  of  the  ratios  for  6  =  10°,  for  example,  we 
read  off  the  coordinates  of  the  extremity  of  the  radius  making 
an  angle  of  10°  with  the  x-axis.  They  are,  approximately, 
X  =  9.8,  and  y  =  1.7,  while  r  =  10.  We  then  have,  approxi- 
mately 


hi 
10 


=  0.17. 


?  =  9_^  =  0.98. 
r      10 


hi 

9.8 


=  0.17. 


e 


17 


The  filling  in  of  the  remainder  of  the  table,  which  is  to  be 
used  in  the  exercises  below,  is  left  as  an  exercise. 

A    method    of    constructing    extensive 

-  tables  of  values  of  these  ratios  is  beyond 

—  the  scope  of  this  course.     But  from  the 
^^  preceding  considerations,  it  should  be  clear 

that,  if  a  value  of  an  angle  6  is  given,  then 
values  of  the  ratios  y/r,  x/r,  and  y/x  are 
determined.  Hence  these  ratios  are  func- 
tions of  6  (Definition,  page  5). 

In  this  chapter  we  shall  study  these 
functions,  their  reciprocals  and  inverses, 
using  the  graphs  of  the  functions  to  tie 

their  properties  together.     We  shall  also  consider  some  of  the 

important  applications^ of  the  functions. 


10** 
20° 
SO** 
40° 
60** 
60* 
70* 
80* 


TRIGONOMETRIC  FUNCTIONS 


163 


EXERCISES 

1.  Find  the  sides  of  a  right  triangle  if  one  acute  angle  is  30°  and  the 
hypotenuse  is  10. 

2.  Find  the  hypotenuse  of  an  isosceles  right  triangle  if  one  side  is  12. 

3.  One  acute  angle  of  a  right  triangle  is  60**  and  the  leg  adjacent  to 
this  angle  is  25.     Find  the  other  leg  and  the  hypotenuse. 

4.  Find  the  side  of  an  equilateral  triangle  whose  altitude  is  8. 

5.  Knd  the  side  of  a  square  if  a  diagonal  is  30. 

6.  A  path  runs  up  the  side  of  a  mountain  at  an  angle  of  40°  to  the 
horizon.  If  a  man  chmbs  along  the  path  for  500  yards,  how  high  will  he 
be  above  the  starting  point? 

7.  A  canal  makes  an  angle  of  20°  with  an  east  and  west  line.  If  a  barge 
moves  at  the  rate  of  4  miles  an  hour,  how  far  will  it  move  east  in  5  hours? 
How  far  north? 

8.  Two  roads  cross  the  canal  in  the  preceding  exercise  at  points  200 
rods  apart,  one  running  east  and  west,  the  other  north  and  south.  How 
large  is  the  triangular  field  bounded  by  the  canal  and  the  roads? 

9.  To  find  the  height  of  a  tree,  a  line  75  feet  long  is  paced  off  from  the 
foot  of  the  tree.  At  the  end  of  the  line  the  angle  subtended  by  the  tree 
is  40°.     How  high  is  the  tree? 

10.  How  high  is  the  sim  (i.e.,  how  many  degrees  above  the  horizon)  if 
a  pole  54  feet  high  casts  a  shadow  20  feet  long? 

11.  A  balloon  is  anchored  by  a  rope  1000  feet  long  which  makes  an  angle 
of  70°  with  the  ground.  How  high  is  the  balloon?  If  a  wrench  happened 
to  drop  from  the  balloon,  how  far  from  the 

point  of  anchorage  would  it  hit  the  ground? 

12.  Find  the  area  of  a  rhombus  whose  side 
is  15  inches  if  one  angle  is  60°. 

13.  What  is  the  length  of  the  edge  of  a  hex- 
agonal nut  that  can  be  cut  from  a  piece  of 
circular  stock  one  inch  in  diameter?  What  is 
the  distance  across  the  flats  (i.e.,  between 
parallel  edges)? 

14.  Holes  are  to  be  drilled  through  a  piece 
of  metal  at  the  vertices  of  a  regular  hexagon, 
3  inches  on  a  side.  The  metal  is  fitted  in  place 
with  a  side  AB  along  the  bed  of  a  milHng 
machine.  What  displacements  of  the  bed  of 
the  machine,  in  the  direction  of  and  perpendicu- 
lar to  AB,  will  bring  the  metal  into  the  proper 
positions  for  drilling  the  holes  in  rotation? 

15.  An  approximate  geometric  method  of  determining  x  is  the  following: 
Draw  the  diameter  AB  of  a  circle  and  the  tangent  CD  at  B.     Construct 


I 


164 


ELEMENTARY  FUNCTIONS 


Fig.  82. 


LBOC  equal  to  30",  and  make  CD  =  3r,  where  r  is  the  radius.    Then  AD 
is  approximately  equal  to  the  semicircumference  ttt. 

Find  to  three  decimal  places  the  approximate  value  of  tt  given  by  this 
construction. 

58.  Angles  of  any  Magnitude.  Let  OX  and  OF  be  two 
lines  drawn  from  an  initial  point  0.  Let  a  line  start  from  co- 
incidence with  OX,  the  initial 
line,  and  rotate  about  0, 
coming  to  rest  finally  in  co- 
incidence with  OP,  the  termi- 
nal line.  The  line  may  rotate 
in  either  direction,  and  it 
may  make  any  number  of 
revolutions  before  coming  to 
rest.  The  Hne  is  said  to 
generate  an  angle  whose  mag- 
nitude is  determined  by  the  amount  and  direction  of  the  rota- 
tion. The  numerical  value  of  the  magnitude  may  be  given  in 
degrees,  right  angles,  or  revolutions. 

The  sign  is  positive  or  negative  according  as  the  direction  of 
rotation  is  counter-clockwise  or  clockwise,  i.e.,  in  the  opposite 
or  in  the  same  direction  as  the  hands  of  a  clock  rotate. 

If  the  terminal  line  of  a  first 
angle  is  the  initial  line  of  a 
second  the  sum  of  the  angles  is 
defined  to  be  the  angle  whose 
initial  line  is  that  of  the  first, 
and  whose  terminal  line  is  that 
of  the  second  angle.  This  is 
analogous  to  the  sum  of  two 
lines  (page  13). 

Two  lines  determine  a  count- 
less nmnber  of  angles.  If  ^  is 
any  one  of  them,  the  others 
differ  from  B  by  an  integral 
multiple  of  360°.  They  may  all  be 
6  +  n360°,   where   w  =  =*=  1,  =^ 


Fio.  83. 


represented   by 


3, 


TRIGONOMETRIC  FUNCTIONS 


165 


f 


The  arcs  in  Fig.  82  indicate  the  three  angles: 

e  =  225°,  6'  =  6  -  360°  =  -  135°,  6"  =  6 -{-  360°  =  585°. 
If  the  initial  Hne  of  an  angle  coincides  with  the  positive 
part  of  the  x-axis,  the  angle  is  said  to  lie  in  the  quadrant  in 
which  the  terminal  Hne  Ues.     Thus  in  Fig.  83,  the  angle  300^ 
lies  in  the  fourth  quadrant,  —  210°  in  the  second. 

The  positive  direetion  on  the  terminal  line  in  such  a  figure  is 
defined  to  be  away  from  the  origin.  For  example,  the  positive 
direction  on  the  terminal  line  of  an  angle  of  180°  is  to  the 
left. 

We  shall  make  an  important  use  of  the  angles  whose  terminal 
lines  bound,  bisect,  or  trisect  the  four  quadrants. 

59.  Trigonometric  Fimctions  of  any  Angle.  Let  6  be  the 
number  of  degrees  in  any  angle  whose  initial  line  coincides 
with  the  positive  part  of  the  x-axis, 
let  P{x,  y)  be  any  point  on  the  termi- 
nal Hne,  and  let  OP  =  r. 

Consider  the  ratio  y/x,  which  is  a 
negative  number  for  the  case  indi- 
cated in  the  figure,  since  x  =  OM  is 
negative  and  y  =  MP  is  positive. 
The  numerical  value  of  y/x  may  be 
found  approximately  by  measuring 
the  lengths  of  MP  and  OM  and  dividing  the  former  by  the 
latter. 

If  P'(x',  2/')  is  any  other  point  on  the  terminal  line,  the  ratios 
2/Vx'  and  y/x  have  the  same  sign,  and  also  the  same  numerical 
value,  since  the  triangles  OMP  and  O'M'P' 
are  similar.  Hence  if  ^  is  given  a  definite 
value,  the  value  of  y/x  is  determined,  and 
therefore  the  ratio  y/x  is  a  function  of  6. 
Similarly,  the  ratio  of  any  one  of  the  num- 
bers X,  y,  r,  to  any  other  is  a  function  of 
the  angle  6.  These  functions  are  called 
trigonometric  functions.  They  are  named 
in  accordance  with  the  definitions  below, 
which  hold  for  Fig.  86  A,  B,  C,  D. 


Fig.  84. 


166 


ELEMENTARY  FUNCTIONS 


Definitions.  If  ^  is  any  angle  whose  initial  line  co- 
incides with  the  positive  part  of  the  x-axis,  if  P{x,  y)  is  any 
point  on  the  terminal  line,  and  if  OP  =  r,  then 

-  =  sine  of  0  =  sin  ^:         -  =  cosecant  d  =  esc  ^ ; 
r  y 


-  =  cosine  of  ^  =  cos  d\     -  =  secant  of  6  =  sec  6; 


-  =  tangent  of  ^  =  tan  ^;  -  =  cotangent  of  ^  =  cot  6. 

To  these  are  sometimes  added 

X 

1  —  =  l-cos^  =  versine  of  ^  =  vers  0, 
r 

l_:2  =  l_sin^  =  coversine  of  ^  =  covers  6, 
r 


M    X 


B  in  quadrant  I. 

yjL 


6  in  quadrant  III. 


Fig.  86. 


0  in  quadrant  IV. 


TRIGONOMETRIC  FUNCTIONS  167 

Since  the  three  ratios  on  the  right  are  the  reciprocals  of  those 
on  the  left,  we  have  the  reciprocal  relations: 

*="*^  =  ter9'  ^^''^^^'  *="'^  =  sT5^-    (^^ 

The  first  table  in  Section  57  gives  the  sine,  cosine,  and  tangent 
of  the  angles  30°,  45°,  and  60°.  The  values  of  the  sines  may 
be  easily  remembered  by  noticing  that  they  are  respectively 
^Vl,  iV2,  §V3,  while  the  cosines  are  the  same  numbers  in  the 
reverse  order.  The  tangent  of  any  one  of  the  angles  may  be  ob- 
tained by  dividing  the  sine  by  the  cosine.     For  we  always  have 

X     x/r     COS  6 

If  d  is  in  quadrant  I,  x,  y,  and  r  are  positive,  and  the  values 
of  the  six  ratios,  or  functions,  are  positive.  But  if  ^  is  in  one  of 
the  other  quadrants,  either  x  or  y,  or  both,  are  negative,  al- 
though r  is  always  positive,  and  hence  these  ratios  may  be  nega- 
tive. For  example,  if  6  is  in  the  second  quadrant,  tan  6  =  y/x 
is  a  negative  number,  since  x  is  negative  and  y  is  positive. 

The  reciprocal  relations  show  that  the  signs  of  cot  6,  sec  0y 
CSC  6,  for  a  given  value  of  6,  agree  respectively  with  the  signs 
of  tan  6,  cos  6,  sin  6.  Hence  it  is  necessary  to  fix  in  mind  the 
signs  of  the  latter  functions  only.  This  is  readily  done  in 
connection  with  the  graphs  (see  Section  61). 

If  the  terminal  Hne  of  6  bounds,  bisects,  or  trisects  one  of 
the  quadrants,  the  functions  of  6  may  be  found  directly  from 
the  definitions,  as  in  the  examples  following,  by  methods  of 
elementary  geometry. 

Example  1.     Find  the  functions  of  0**. 

Let  P{x,  y)  be  any  point  on  the  terminal  line,  which  coincides  with  the 
positive  part  of  the  x-axis,  since  B  =  0.    Then 
2/  =  0,  and  r  =  x.    Hence,  by  the  definitions,  V* ' 

sin  0°  =  y/r  =  0/r  =  0. 
cos  0°  =  x/r  =  x/x  =  1. 


tanO**  =  2//x  =  0/a;  =  0.  ^ 


— o >■ 

M  X 


sec  0°  =  r/x  =  x/x  =  1. 
But  the  definition  of  cot  6  involves  divi- 
sion by  y,  and  as  !/  =  0  if  ^  =  0,  co<  0°  does  ^^°'  ^'^• 
not  exist.    However,  cot  6  is  the  reciprocal  of  tan  6,  by  (1).    Aa  B  ap- 


168 


ELEMENTARY  FUNCTIONS 


Fig.  88. 


preaches  zero,  tan  6  approaches  zero,  and  hence  cot  6  becomes  infinite 
as  6  approaches  zero  (IV,  page  118).  This  is  sometimes  indicated  by  the 
symbols  cot  0°  =  <». 

In  like  manner,  as  d  approaches  zero,  esc  6  becomes  infinite. 
Example  2.     Find  the  functions  of  210°. 

The  acute  angles  of  the  triangles  0PM  are  30°  and  60°.  Hence  the 
numerical  value  of  OP  is  twice  that  oi  MP.    If  we  let  the  numerical  value 

of  MP  be  1,  then  that  of  OP  is  2,  and  hence 
that  of  OP  is  Vs.     But  the  coordinates  of 
P  are  negative.     Hence  we  may  take 
X  =  -V3,     2/  =  -  1,    r  =  2. 
Then  by  the  definitions 
sm  210°  =  y/r  =  -1/2. 
CSC  210°  =  r/y  =  2/(- 1)  =  -  2. 
cos  210°  =  x/r  =  -\/3/2^ 
sec  210°  =  r/x  =  2/(_-V3)  =  -  2V3/3. 
tan  210°  =  y/x  =  -  l/(-  V3)  =  V3/3. 
cot  210°  =  x/y  =  -  V3/(-  1)  =  V3. 
Since  the  functions  of  two  angles  with  the  same  terminal  line 
may  be  defined  by  means  of  the  same  triangle,  and  since  all 
angles  with  the  same  terminal  line  may  be  expressed  in  the 
form    ^  +  n360°    (Section   58),   we 
have  the 

Theorem.  The  trigonometric  func- 
tions of  an  angle  are  unchanged  if 
the  angle  is  increased  or  decreased  hy 
an  integral  multiple  of  360°.  Syrrir 
bolically, 

sin  (6  +  n360°)  =  sin  6, 
cos  (d  +  n360°)  =  cos  dy 
where  n  =  ±  1,  =*=  2,  =fc  3,  etc. 

Definition.  A  function  is  said  to  be  periodic  if  its  value 
is  unchanged  when  the  value  of  the  variable  is  increased  by  a 
constant,  that  is,  if  fix  +  c)  =f{x).  If  c  is  the  smallest  con- 
stant of  this  sort,  it  is  called  the  period  of  the  function. 

For  example,  the  height  of  the  tide  at  the  seashore  is  a 
periodic  function  of  the  time.  For  if  approximately  12  hours 
and  25  minutes  are  added  to  the  time,  the  height  of  the  tide 
will  be  the  same. 


TRIGONOMETRIC  FUNCTIONS 


169 


The  theorem  shows  that  the  trigonometric  Junctions  are 
periodic,  for  if  0  is  increased  by  360°  the  values  of  the  func- 
tions are  unchanged.  It  will  appear  later  that  360°  is  the 
period  of  the  sine  and  cosine,  and  their  reciprocals,  while 
180°  is  the  period  of  the  tangent  and  cotangent.  This  is  a 
characteristic  property  of  the  trigonometric  functions,  which 
distinguishes  them  from  all  algebraic  functions. 


EXERCISES 

1.  Find  all  the  functions  of  each  of  the  so-called  quadrantal  angles: 
(a)  0°;   (b)  90°;   (c)  180°;  (d)  270°;   (e)  360°. 

2.  Find  all  the  functions  of  the  angles: 

(a)  135°.  (b)  330°.  (c)  -  45°.  (d)  240°.  (e)   -  210°. 

(f)   480°.  (g)  315^  (h)  -  150°.  (i)   300°.  (j)       990°. 

3.  What  positive  angles  less  than  360°  have  the  same  functions  as 
(a)  540°,  (b)   -  60°,  (c)  1320°,  (d)   -  675°,  (e)  653°? 

4.  Determine  the  sign  of: 


(a)  cos  B,  6  in  quadrant  II. 
(c)  cos  6,  6  in  quadrant  IV. 
(e)  sin  6,  6  in  quadrant  IV. 
(g)  cot  6,  6  in  quadrant  IV. 


(b)  tan  9,  9  in  quadrant  III. 

(d)  tan  9,  9  in  quadrant  II. 

(f)  sec  9,  9  in  quadrant  III. 

(h)  CSC  9,  9  in  quadrant  II. 


120' 


p 


6.   Determine  the  sign  of  each  of  the  functions  in  each  of  the  quadrants. 
Tabulate  the  results. 

6.  Build  a  table  of  values  of  sin  9  for  9  =  0°,  30°,  45°,  60°,  90= 
135°,  150°,  180°,  210°,  225°,  240°,  270°,  315°,  300°,  330°,  360°. 

7.  Build  a  table  of  values, 
for  the  angles  given  in  Exer-  '^  ^^ 
cise  6,  for  (a)  cos  9;  (b)  tan  9] 
(c)  cot  9)  (d)  sec  9]  (e)  esc  9. 

8.  Construct  all  the  posi- 
tive angles  less  than  360°  for 
which  sin  0  =  -  f . 

Since  sin  9  =  y/r,  the 
problem  is  equivalent  to 
constructing  right  triangles 
with  hypotenuse  equal  to  5, 
and  vertical  side  equal  to  3 
and  lying  below  the  x-axis. 
Hence,  describe  the  circle 
with  center  0  and  radius  5,  and  draw  the  line  parallel  to  the  x-axis  and 
3  units  below  it.     Let  their  intersections  be  P  and  P'.     Then  the  lines 


Fig.  90. 


170  ELEMENTARY  FUNCTIONS 

OP  and  OP'  are  the  terminal  lines  of  the  required  angles.     How  many 
such  angles  are  there?    How  many  positive  and  less  than  360°? 

9.  By  the  method  of  the  preceding  exercise,  construct  all  the  positive 
angles  less  than  360°  for  which 

(a)  cos  6  =  \.  (b)  tan  0  =  -  |.  (c)  esc  0  =  3. 

(d)  sin  0  =  4.  (e)  cos  0  =  -  0.3  (f)  cot  0  =  2. 

10.  Find  all  the  functions  of  6  given : 

(a)  cos  0  =  -  i^j,  and  0  in  quadrant  II. 

(b)  tan  0  =  1,  and  0  in  quadrant  III. 

(c)  sin  0  =  -  H>  and  0  in  quadrant  IV. 

(d)  cot  0  =  2,  and  0  in  quadrant  I. 

(e)  sin  0  =  t,  and  0  in  quadrant  II. 

(f)  sec  0  =  ^-,  and  0  in  the  fourth  quadrant. 

11.  The  intensity  of  light,  /,  varies  inversely  as  the  square  of  the  dis- 
tance, d,  from  the  source.  If  a  street  light  is  at  the  top  of  a  concrete  post 
10  feet  high,  AB,  express  /  at  a  point  C  on  the  pavement  as  a  function  of 
0  =  AACB. 

12.  An  aeroplane  rises  along  a  straight  line,  which  makes  an  angle  0 
with  a  straight  road  directly  below  the  path  of  the  aeroplane,  at  the  rate 
of  50  miles  an  hour.  Express  as  a  function  of  0  the  speed  at  which  an 
automobile  must  move  along  the  road  to  keep  under  the  aeroplane. 

13.  If  the  hypotenuse  of  a  right  triangle  is  10,  express  the  area  as  a 
function  of  one  of  the  acute  angles. 

14.  Construct  the  path  of  a  point  on  the  rim  of  a  wheel  rolling  on  a 
level  road,  by  rolling  a  coin  along  a  ruler,  without  slipping,  and  marking 
a  number  of  positions  of  a  point  on  the  edge  of  the  coin.  This  curve  de- 
fines y,  the  height  of  the  moving  point  above  the  road,  as  a  function  of  x, 

the  distance  measured  along  the  road. 
Show  that  this  function  is  periodic,  and  find 
its  period.  What  can  be  said  of  the  graph 
of  a  periodic  function? 

60.  Radians.  To  find  the  number 
of  degrees  in  the  angle  u  subtended 
at  the  center  of  a  circle  of  radius  r 
by  an  arc  whose  length  is  r,  we  com- 
pare the  angle  with  the  complete 
angle  about  the  center.  Since  angles 
at  the  center  are  proportional  to  the 

subtending  arcs,  and  since  an  angle  of  360°  is  subtended  by  the 

entire  circumference,  we  have 


TRIGONOMETRIC  FUNCTIONS  171 

J£ ^  /.x 

360      27rr  ^  ^ 
and  hence 

u  =  180/7r  =  180/3.1416  =  57°.295.  (2) 

Hence  this  angle  does  not  depend  on  r,  but  is  the  same  for 
all  circles,  and  it  may  therefore  be  used  as  a  unit  angle. 

Definition.  A  radian  is  the  angle  subtended  at  the  center 
of  a  circle  by  an  arc  whose  length  is  equal  to  the  radius. 

Equation  (2)  enables  us  to  reduce  radians  to  degrees.    The 

most  convenient  form  of  this  equation  is  the  important  relation 

w  radians  =  180  degrees.  (3) 

From  this  we  have,  for  example,  that  90°  =  ^72  radians, 
and  it  is  customary  to  speak  of  7r/2  radians  rather  than 
3.1416/2  =  1.5708  radians.     Similarly, 

It  is  customary  to  express  in  terms  of  ir  the  number  of  radians 
in  any  angle  which  is  a  simple  multiple  or  submultiple  of  180°. 

When  the  degree  or  the  right  angle  is  used  as  the  unit  that 
fact  is  usually  indicated.  Thus  we  write  6  =  180°,  or  6  =  2 
rt.  A .  But  if  the  unit  is  the  radian  we  merely  write  6  =  w  with- 
out indicating  the  unit. 

The  number  of  radians  in  an  angle  is  called  its  circular  measure. 

It  is  customary  to  use  the  radian  as  the  unit  angle  in  draw- 
ing the  graphs  of  the  trigonometric  functions  (see  the  follow- 
ing section).  Another  elementary  use 
is  given  by  the 

Theorem.  The  length  of  an  arc  of 
a  circle  is  equal  to  the  radius  multi- 
plied by  the  number  of  radians  in 
the  angle  subtended  at  the  center. 
Symbolically, 

Arc  il5  =  r  X  /.AOB  {m  radians). 

For  if  A B  is  any  arc,  if  ^  is  the  p^^  92 

circular    measure   of    the   subtended 
angle  AOB,  and  if  arc  AC  =  r,  so  that  /.AOC  =  1,  we  have 

=  7'  whence  AB  =  rd, 

r        1 


172  ELEMENTARY  FUNCTIONS 

The  radian  is  the  unit  used  in  all  theoretical  work  in  the 
calculus  and  higher  mathematics. 

Tables  for  converting  radians  into  degrees,  and  degrees  into 
radians,  are  to  be  found  on  page  32  of  Huntington's  Tables. 


EXERCISES 

1.  Reduce  the  following  angles  to  radians: 

(a)  The  quadrantal  angles,  namely,  0°,  90"*,  180*,  270*,  360*. 

(b)  The  angles  whose  terminal  lines  bisect  the  quadrants,  namely,  45*, 
135*,  225°,  315°. 

(c)  The  angles  whose  terminal  lines  trisect  the  quadrants,  namely, 
30°,  60°,  120°,  150°,  210°,  240°,  300°,  330*. 

2.  Reduce  the  following  angles  to  degrees: 

(a)  7r/6,  37r,  77r/2,  27r/3. 

(b)  27r/3,  7r/4,  Stt/S,  57r/4,  ll7r/3. 

3.  Using  Huntington's  Tables,  page  32,  express  the  following  angles  in 
radians,  as  decimal  fractions: 

(a)  10°,  75°,  110°,  340°,  15°.34,  3°.77. 

(b)  The  angles  in  Exercise  1. 

4.  Using  Huntington's  Tables,  express  the  following  angles  in  degrees: 
0.25,  1.13,  1.465,  0.8327,  3.2476. 

5.  If  the  radius  of  a  circle  is  10  inches,  find  the  angle  subtended  at  the 
center  by  an  arc  15  inches  long. 

6.  If  the  radius  of  a  circle  is  4  inches,  find  the  length  of  an  arc  which 
subtends  an  angle  of  173*  at  the  center. 

7.  A  strip  of  tin  8  inches  wide  is  bent  into  the  form  of  a  trough  whose 
cross  section  is  an  arc  of  a  circle.  Express  the  angle  subtended  at  the 
center  as  a  function  of  the  radius. 

8.  Find  the  angle  at  the  center  of  the  earth  subtended  by  an  arc  of  the 
equator  one  mile  long.  In  doing  this,  which  one  of  the  radii  of  the  earth 
given  on  the  inside  of  the  back  cover  of  Huntington's  Tables  should  be 
used? 

9.  If  the  radius  of  a  circle  is  12  inches,  find  the  length  of  an  arc  subtend- 
ing a  central  angle  of  27r/3,  and  the  area  of  the  sector  bounded  by  the  arc 
and  the  radii  drawn  to  its  extremities. 

10.  Show  that  the  area  of  a  sector  of  a  circle  of  radius  r  is  §  r^O,  where 
6  is  the  number  of  radians  in  the  central  angle  of  the  sector.  Hint.  The 
area  of  a  sector  is  one-half  the  product  of  the  radius  and  the  arc  of  the 
sector. 

61.  Graphs  of  the  Trigonometric  Functions.    In  construct- 


TRIGONOMETRIC  FUNCTIONS 


173 


ing  these  graphs  it  is  customary  to  use  the  same  unit  on  both 
axes,  and  to  measure  6  in  radians. 

Graph  of  sin  6,  A  sufficiently  extensive  table  of  values  is 
obtained  by  taking  the  values  of  6  for  which  the  terminal  Une 
bounds,  bisects,  or  trisects  one  of  the  quadrants.  The  values 
of  sin  d  for  the  angles  30°  =  7r/6,  45°  =  7r/4,  60°  =  7r/3  are 
given  in  the  first  table  in  Section  57,  and  for  the  angles  0°  and 
210°  =  77r/6  in  the  examples  in  Section  59.  The  table  of  values 
below  is  identical  with  that  asked  for  in  Exercise  6,  follow- 
ing Section  59,  except  that  the  angles  are  now  expressed  in 
radians. 

To  plot  the  pairs  of  values  in  the  table,  choose  a  convenient 
unit  on  the  vertical  axis,  and  on  the  ^-axis  lay  off  tt  =  3-f  units 
and  27r  =  6f  units.  The  points  B  =  7r/2  and  B  =  37r/2  are 
obtained  by  bisecting  these  segments,  giving  four  segments  on 
the  ^-axis  corresponding  to  the  four  quadrants.  The  remain- 
ing abscissas  may  be  constructed  by  bisecting  and  trisecting 
these  segments.  Having  determined  the  points  on  the  ^-axis 
corresponding  to  the  values  of  B  in  the  table,  ordinates  are 
erected  equal  to  the  respective  values  of  sin  B,  and  the  curve  is 
then  drawn. 


Fig.  93. 


sin  6 


0,   7r/6,    7r/4,    7r/3,   7r/2,   27r/3,   37r/4,    5t/6,    x,   77r/6,   57r/4 


0,    0.5,    0.7,    0.9,      1,      0.9,      0.7, 

e     !  47r/3,  37r/2,  57r/3,   77r/4, 


0.5,      0,-0.5,-0.7 

ll7r/6,    2x 


^1   -0.9, 


1,    -0.9, 


0.7, 


0.5, 


Periodicity  of  sin  B.  If  the  terminal  hne  of  B  starts  in  co- 
incidence with  the  positive  part  of  the  a;-axis  (Section  58), 
the  angle  B  may  have  any  one  of  the  values: 

.  .  .  -  47r,  -  27r,  0,  27r,  47r,  .  .  . 


174  ELEMENTARY  FUNCTIONS 

As  the  terminal  line  makes  a  complete  revolution,  6  may  in- 
crease through  any  one  of  the  intervals: 

.  .  .  -  47r  to  -  27r,    -  2w  to  0,    0  to  27r,    2t  to  47r,  .  .  . 

But  no  matter  through  which  of  these  intervals  6  increases, 
sin  6  will  vary  through  the  same  set  of  values  (Theorem, 
Section  59).  Hence  the  part  of  the  graph  in  each  interval  is 
congruent  to  that  in  the  interval  from  0  to  27r,  which  was 
plotted  above. 

Starting  at  the  origin,  the  graph  does  not  begin  to  repeat  until 
0  =  2w.     Hence  27r,  or  360°,  is  the  smallest  constant  such  that 

sin  (6  -h  27r)  =  sin  6, 

and  hence  27r,  or  360°  is  the  period  (Definition,  Section  59)  of 
sin  6. 

The  part  of  the  graph  from  0  to  2t  should  be  fixed  in  mind 
carefully,  and  the  portion  corresponding  to  each  quadrant  noted, 
as  the  graph  affords  a  simple  means  of  remembering  the  following 
properties  of  sin  6  (see  page  42). 

Zeros  of  sin  6.  For  angles  less  than  27r,  sin  ^  =  0  if  ^  =  0 
or  TT. 

From  the  periodicity  of  the  function,  the  other  zeros  are 
therefore 

0  +  2n7r  =  =*=  27r,     ±  47r,      =»=  Qir,  .  .  . 
and  TT  +  2n7r  =  =t  Stt,     =t  5t,     ±  Ttt,  .  .  . 

Sign  of  sin  6.  Sin  6  is  positive  if  6  is  in  the  first  or  second 
quadrant,  negative  if  6  is  in  the  third  or  fourth  quadrant. 
The  periodicity  shows  that  this  holds  whether  or  not  6  is  posi- 
tive and  less  than  27r. 

Maximum  and  minimum  values  of  sin  6.  The  maximum 
value  between  0  and  27r  is  sin  (7r/2)  =  1,  and  the  minimum 
value  sin  (37r/2)  =  -  1.  The  other  values  of  6  for  which  sin 
^  =  =b  1  may  be  found  by  means  of  the  periodicity. 

Changes  of  sin  6. 

As  6  increases  from  0  to  7r/2,  sin  6  increases  from  0  to  1. 

As  6  increases  from  ir  /2  to  tt,  sin  6  decreases  from  1  to  0. 

As  6  increases  from  t  to  37r/2,  sin  6  decreases  from  0  to  -  1. 


TRIGONOMETRIC  FUNCTIONS 


175 


As  6  increases  from  Stt  /2  to  27r,  sin  6  increases  from  —  1  to  0. 

Notice  that  the  numerical  value  of  sin  6  cannot  exceed  unity, 
and,  in  particular,  that  it  does  not  become  infinite. 

Symmetry  of  the  graph.     The  graph  appears  to  be  synmietrical 
with  respect  to  the  origin.    Hence,  probably, 
sin  {-  6)  =  -  sin  6. 

This  may  be  proved  as  follows : 
Construct  any  angle  6,  and  then 
the  angle  -  6.  On  their  termi- 
nal Hnes  take  OP  =  O'P',  so  that 
r  =  r' .  Then  since  P'  is  sym- 
metrical to  P  with  respect  to  the 
a;-axis  (why?),  we  have  x  =  x' 
and  y  =  -  y'.    Then 


sin  (■ 


0)  =  ^  = 

r 


—  sm 


6, 


Fig.  94. 


The  graphs  of  cos  6  (Fig.  95)  and  tan  6  (Fig.  96)  are  con- 
structed in  hke  manner,  and  should  be  fixed  in  mind. 

The  various  properties  of  reciprocal  functions  (page  118) 
enable  us  to  get  certain  properties  of  cot  d,  sec  d,  and  esc  6 
directly  from  the  graphs  of  tan  6,  cos  6,  and  sin  d  respectively. 
(See  also  Exercise  2  below.) 


cos  d 


I 


Fig.  95. 


6     1  0,  7r/6,  7r/4,  x/3,  7r/2,  27r/3,  37r/4,  57r/6,   tt,  77r/6,  57r/4 


1,  0.9,  0.7,  0.5, 

d      I  47r/3, 
cos  d  \    -0.5, 


0,  -0.5,  -0.7,  -0.9,  -  1,  -0.9, 

37r/2,  57r/3,  77r/4,  ll7r/6,  27r 


0.7 


0,   0.5,   0.7, 


0.9, 


176 


ELEMENTARY   FUNCTIONS 


e 


tan  d 


Fig.  96. 

0,  7r/6,  7r/4,  7r/3,  7r/2,  27r/3,  37r/4,  Stt/G,  tt,  Ttt/G,  5t/4 


0,  0.6,  1,  1.7,   00,  -1.7,  -1,  -0.6,  0,  0.6, 

e      I  47r/3,    37r/2,     57r/3,    77r/4,    IItt/G,    27r 
tan  ^  I      1.7,       00,      -1.7,       -1,    -0.6,     0 


EXERCISES 

1.  Discuss  the  periodicity,  zeros,  values  of  6  for  which  the  function 
becomes  infinite,  sign,  maxima  and  minima,  changes  and  symmetry  of 
cos  6  and  tan  d. 

2.  Sketch  on  the  same  axes  the  graphs  of  the  pairs  of  functions  follow- 
ing, and  discuss  the  second  function  with  respect  to  the  properties  listed 
in  Exercise  1. 

(a)  sin  6  and  esc  B.        (b)  cos  B  and  sec  B.       (c)  tan  B  and  cot  B. 

3.  Construct  the  graphs  of  the  six  trigonometric  functions  on  the  same 
axes. 

4.  What  properties  of  the  functions  can  be  inferred  from  graphs  or 
the  same  axes  of 

(a)  sin  B  and  cos  0?  (b)  tan  B  and  cot  ^?  (c)  sec  B  and  esc  Bl 

6.  Describe  the  motion  of  a  particle  on  a  straight  line  if  its  distance  s 
from  a  fixed  point  on  the  line  at  any  time  t  is  given  by  s  =  sin  t.  Does 
such  a  motion  approximate  any  motion  occurring  in  nature? 

6.  On  the  same  axes  sketch  the  graphs  of  the  fimctions: 


TRIGONOMETRIC  FUNCTIONS 


177 


(a)  sin  6  and  2  sin  6.  (b)  cos  6  and  3  cos  d. 

(c)  tan  $  and  i  tan  ^.  (d)  cot  6  and  0.2  cot  6. 

(e)  sec  ^  and  ^  sec  ^.  (f)    esc  6  and  |  esc  0. 

7.  Construct  and  discuss  the  graph  of 

(a)  vers  ^  =  1  -  cos  d.  (b)  covers  6  =  1  -  Bin  6. 

8.  By  the  addition  of  ordinates  (see  Exercise  4,  page  44)  construct  the 
graph  of 

(a)  sin  X  +  cos  x.  (b)  2  sin  x  -  cos  x.  (e)  sin  x  +  3  cos  x. 

62.  Functions  of  Complementary  Angles.  Construct  an 
acute  angle  6  and  its  complement  90°  —  6  with  their  initial 
lines  coinciding  with  the  positive  part  of  the  x-axis.  On  their 
terminal  lines  take  OP  =  0T\  so  that  r  =  r'.  Then  P  and  P' 
are  symmetrical  with  respect  to  OA, 
the  bisector  of  the  first  quadrant 
(why?),  and  hence  x  =  y'  and  y  =  x\ 

Then 


sin  (90°  -  0)  = 

and 

cos  (90°  -  6)  = 


y 


-,  =  cos  0 

r 


(1) 


sin  6.    (2) 


Using  (2),  page  167,  with  (1)  and 
(2),  we  then  have 

sin  (90°  -  6) 


tan  (90°-  d) 


cos  (90°  -  0) 

1 


Also,  cot  (90°  -   6)   =   —      .^o         n^    -        .    n 

'        ^  ^      tan  (90    -  6)      cot  6 

and  in  like  manner, 

sec  (90°  -  ^)  =  CSC  6,  esc  (90°  -  6)  =  sec  d.  (5) 

The  cosine,  cotangent  and  cosecant  of  an  angle  are  so  named 
because  they  are  respectively  the  sine,  tangent,  and  secant  of 
the  complementary  angle,  as  is  shown  by  these  relations.  The 
former  functions  are  called  the  cofunctions  of  the  latter,  re- 
spectively, and  vice  versa.  With  this  terminology,  the  six 
relations  (1)  to  (5)  may  be  stated  as  the 


178 


ELEMENTARY  FUNCTIONS 


Theorem.  The  functions  of  any  acute  angle  are  equal  re- 
spectively  to  the  cofunctions  of  the  complementary  angle. 

A  method  of  extending  the  proofs  of  these  relations  for  any 
value  of  Of  not  necessarily  acute,  will  be  given  in  Section  68. 

63.  Tables  of  Trigonometric  Functions.  As  a  consequence  of 
the  theorem  in  the  preceding  section,  tables  of  values  of  the 
trigonometric  functions  may  be  printed  in  very  compact  form. 

Since  cos  (90°  —  6)  =  sin  6,  a  table  of  sines  of  any  set  of 
angles  is  also  a  table  of  cosines  of  the  complementary  angles. 
The  complements  of  0°,1°,  2°,  .  .  .,  88°,  89°,  90°  are  respectively 
90°,  89°,  88°,  .  .  .,  2°,  1°,  0°.     Hence: 

1.  The  table  of  sines  on  pages  8  and  9  of  Huntington's 
Tables  is  also  a  table  of  cosines  if  read  backward. 

2.  The  sines  and  cosines  of  angles  from  0°  to  45°  in  the 
Condensed  Table  on  the  inside  of  the  back  cover  of  the  Tables 
are,  if  read  upward,  the  cosines  and  sines  respectively  of  the 
angles  from  45°  to  90°. 

In  like  manner,  tan  6  and  cot  B  may  be  given  in  one  table, 
and  so  also  may  sec  6  and  esc  6. 

Thus  the  theorem  on  functions  of  complementary  angles 
makes  it  practicable  to  reduce  by  one-half  the  space  devoted 
to  a  table  of  trigonometric  functions. 

Fractional  parts  of  a  degree,  in  Huntington's  Tables,  are 
given  in  tenths  and  hundredths  instead  of  in  minutes  and 
seconds.  One  of  the  merits  of  this  decimal  method  of  sub- 
division is  that  the  process  of  interpolation,  in  finding  a  func- 
tion of  6,  is  identical  with  that 
used  earher  in  the  other  tables 
(page  121). 

Notice  that,  in  the  body  of 
the  tables,  the  decimal  point, 
and  any  figures  preceding  it, 
are  usually  printed  only  in  the 
first  column.  For  example,  tan 
53°.32  =  1.3426. 
^'°-  ^^-  The  process  of  finding  6,  if 

a  function  of  6  is  given,  is  illustrated  in  the  examples  following. 


J^ 

F^^ 

--^ 

D 

7?  -""^ 

I 

i 

1 

H 

1 

d 

A 

E 

C 

^ 

TRIGONOMETRIC  FUNCTIONS  179 

Example    1.     Find  d  if  sin0  =  0.4321. 

Searching  through  the  body  of  the  table  of  sines,  we  find  that  0.4321  is 
given  in  the  table,  and  reference  to  the  margin  shows  that  6  =  25°.6. 

Example  2.    Find  ^  if  sin  ^  =  0.4332. 

A  search  in  the  body  of  the  table  of  sines  shows  that  0.4332  lies  between 
0.4321  and  0.4337,  which  are  the  sines  of  25°.6  and  25''.7.  Fig.  98 
shows  the  graph  of  sin  6  between  these  angles,  on  the  assumption  that  it 
is  straight  (compare  the  assumption  on  page  122).  To  find  6,  graphically, 
is  to  find  the  value  of  ^  at  J^J  if  EF  =  0.4332. 

The  slope  of  the  graph  is  the  difference  of  the  ordinates  of  B  and  D, 
HD  =  0.0016,  divided  by  the  difference  of  the  abscissas,  ^C  =  0.1  (Defini- 
tion, page  50).  It  is  also  equal  to  the  difference  of  the  ordinates  of  F 
and  D,  ID  =  0.0005,   divided  by   the   difference  of  the  abscissas,   EC. 

Hence  0.0016  _  0.0005 

AC     ~     EC    ' 

whence  ^^  ^  0^  ^^  =  4  AC  =  0.3AC  =  0.3  x  0.1  =  0.03. 

0.0016  16 

Hence  0,1  E,  d  =  25^.67. 

The  arithmetical  processes  used  in  interpolating  to  find  an 
angle  of  which  a  function  is  given  are: 

Find  the  two  successive  numbers  in  the  proper  table  between 
which  the  given  number  lies. 

Find  the  difference  between  the  given  number  and  that  one  of 
these  two  numbers  to  which  it  is  nearer,  and  divide  this  difference 
by  the  tabular  difference. 

Apply  the  result  as  a  correction  to  the  last  digit  of  the  angle 
whose  function  was  used  in  getting  the  difference  above,  in  such  a 
way  that  the  result  lies  between  the  two  angles  corresponding  to 
the  two  numbers  in  the  table. 

Example  3.     Find  6  if  cos  6  =  0.4815. 

Searching  through  the  body  of  the  table,  we  find  that  0.4815  lies  between 
0.4818  and  0.4802,  the  cosines  of  61*^.2  and  61°.3.  It  is  nearer  the  former, 
the  difference  being  3,  neglecting  the  decimal  point,  while  the  tabular 
difference  is  16.  The  correction  to  be  applied  to  61°.2  is  therefore  x%  of 
one-tenth  of  a  degree,  so  that  d  =  QV^j^  =  61'*.22. 

In  using  the  table  of  tenths  of  the  tabular  difference,  in  the  margin, 
find  the  difference  3  as  above.  Then  look  for  3  in  the  margin.  It  lies  in 
the  column  headed  2.  Hence  3  is  2-tenths  of  the  tabular  difference,  and 
the  digit  2  is  annexed  to  61*'.2,  giving  61°.22. 

The  use  of  the  tables  for  other  than  acute  angles  will  be  considered  in 
Section  69. 


180  ELEMENTARY  FUNCTIONS 


EXERCISES 


1.  Find  the  cosine,  cotangent,  and  cosecant  of  72° A3.  Find  the  sine, 
tangent,  and  secant  of  17°.57.     Compare  the  two  sets  of  results. 

2.  Find  the  six  functions  of  73°.26,  and  verify  the  fact  that  the  last 
three  functions  are  the  reciprocals  of  the  first  three  in  the  reverse  order. 

3.  Find  the  acute  value  of  6,  illustrating  the  interpolation  graphically, 
if 

(a)  sin  e  =  0.9235.  (d)  cot  6  =  0.3603. 

(b)  cos  6  =  0.3129.  (e)  sec  6  =  1.5165. 

(c)  tan  ^  =  1. 1603.  (f)   esc  d  =  5.515. 

4.  The  arithmetical  operations  in  this  exercise  should  be  performed 
mentally.     Find  the  acute  value  of  6  such  that 

(a)  tan  6  =  0.8134;    tan  6  =  1.8134;     cos  6  =  0.3019. 

(b)  sin  e  =  0.8436;    sec  d  =  4.057;       cot  6  =  0.6142. 

(c)  cos  d  =  0.0589;    esc  6  =  1.6397;      sin  6  =  0.16543. 

(d)  cos  d  =  0.0630;    cos  6  =  0.06300;    tan  d  =  1.4628. 

5.  Construct  the  line  through  the  origin  whose  slope  is  2,  and  find  the 
angle  between  the  line  and  the  a;-axis. 

6.  Construct  a  table  of  values  of  0  and  sin  6  for  values  of  $  taken  every 
15°  from  0°  to  90**,  expressing  6  in  radians  decimally  instead  of  in  tenns  of 
TT  (see  Tables,  page  32),  and  giving  the  values  of  6  and  sin  6  to  two  decimal 
places.  Construct  the  graph  as  accurately  as  possible  from  this  table, 
using  a  large  scale.  On  the  same  axes  draw  the  graph  of  6.  What  ap- 
proximate value  of  sin  0  for  very  small  angles  is  suggested  by  these  graphs? 
Using  the  Condensed  Tables  on  the  inside  of  the  back  cover  of  the  Tables, 
determine  for  how  large  an  angle  this  approximation  is  correct  to  three 
decimal  places;  to  four  decimal  places.  What  is  the  limit  of  sin  0/0  as 
6  approaches  zero? 

7.  Solve  Exercise  6  replacing  sin  0  by  tan  0. 

8.  Using  the  properties  of  the  functions  sin  0,  cos  0  and  tan  0  suggested 
by  the  symmetry  of  their  graphs,  find  the  sine,  cosine,  and  tangent  of  the 
negative  angles,  -  24**.32,  -  48**.27,  -  68**.46. 

64.  Solution  of  Right  Triangles.  It  is 
customary  to  denote  the  magnitude  of 
the  angles  of  a  triangle  ABC  hy  A,  B,  C 
and  the  lengths  of  the  sides  opposite  by 
a,  6,  c  respectively.  In  a  right  triangle, 
the  right  angle  is  usually  denoted  by  C, 
and  the  hypotenuse  by  c.  A  right  tri- 
Fio.  99.  ^^gjg  j^^Q  jjjg^y  bg  placed  with  reference 

to  coordinate  axes  so  that  A  is  at  the  origin,  AC  lies  along 


y 

^ 

C/^ 

1 

3 

a 

0 

A 

b 

C 

/   X 

TRIGONOMETRIC  FUNCTIONS  181 

the  positive  a;-axis,  and   the  hypotenuse  AB  Ues  in  the  first 
quadrant.     Then  the  definitions  on  page  166  show  that 

sin  A  =  a /c,  cos  A  =6 /c,  tan  A  =  a/b)  ^  . 

cot  A  =  b/ttf  sec  A  =  c/b,  CSC  A  =  c/aj  ^  ^ 

It  is  not  always  convenient  to  place  the  triangle  on  coordi- 
nate axes,  and  sometimes  other  letters  must  be  used,  and  hence 
it  is  desirable  to  remember  these  formulas  in  words,  as  follows : 

The  sine  of  an  acute  angle  of  a  right  triangle  is  the  ratio  of 
the  side  opposite  the  angle  to  the  hypotenuse. 

The  cosine  of  an  acute  angle  of  a  right  triangle  is  the  ratio  of 
the  side  adjacent  to  the  angle  to  the  hypotenuse. 

The  tangent  of  an  acute  angle  of  a  right  triangle  is  the  ratio 
of  the  side  opposite  the  angle  to  the  adjacent  side. 

Analogous  statements  for  the  last  three  functions  are  readily 
obtained  by  the  reciprocal  relations  (page  166). 

The  sides  and  angles  of  a  triangle  are  called  its  parts.  In 
order  to  construct  a  right  triangle,  we  must  be  given  two  parts, 
in  addition  to  the  right  angle,  of  which  at  least  one  must  be  a 
side.  To  solve  a  triangle  is  to  find  the  unknown  parts  from  the 
known. 

Every  right  triangle  may  be  solved  by  means  of  formulas  (1)  and 
the  fact  that  A  -\-  B  =  90°.  But  the  Pythagorean  Theorem  is 
sometimes  convenient. 

In  solving  right  triangles  but  two  essentially  different  cases 
arise: 

f  I.  Given  a  side  and  an  angle,  to  find  the  other  two  sides  use 
those  two  of  equations  (1)  which  contain  the  unknown  sides  in 
the  numerators  and  the  given  side  in  the  denominator. 

II.  Given  two  sides,  to  find  an  angle,  use  that  one  of  the  two 
equations  (1)  containing  the  given  sides  which  leads  to  the 
simpler  division,  and  then  to  find  the  third  side  use  either  of 
the  two  equations  containing  the  third  side  in  the  numerator. 

In  either  case  the  second  angle  is  found  from  A  +  B  =  90°. 

The  use  of  the  equations  indicated  in  these  rules  will  lead 
to  multiplication,  and  division  by  a  value  of  sin  6,  for  ex- 
ample, is  avoided. 


182 


ELEMENTARY  FUNCTIONS 


It  is  desirable  that  figures  be  constructed  accurately,  as 
the  figure  may  show  the  absurdity  of  an  incorrect  result.  A 
good  check  on  the  accuracy  of  the  computation  may  be  ob- 
tained by  constructing  the  triangle  with  scale  and  protractor, 
and  measuring  the  unknown  parts. 

Example  1.     Solve  the  right  triangle  A  =  37°.24,  b  =  9. 
Solution.    5  =  90**  -  A  =  52°.76. 


tan  A ,  whence  a 


h  tan  A . 
9  X  0.7601 
6.8409. 


6=9 
Fig.  100. 


=  sec  A,  whence  c  =  6  sec  A 

=  9  X  1.2561 
=  11.3049. 


Check.    The  accuracy  of   the  computation 
may  be  checked  by  finding  b  from  a  and  c.     By 
the  Pythagorean  Theorem,  and  tables  of  squares  and  square  roots, 

VSLO" 


6  -  Vc2  -  a2  =  \/127.7  -  46.7 


9, 


which  agrees  with  the  given  value  of  b. 

Example  2.     Solve  the  right  triangle  a  =  7,  &  =  9. 

The  given  sides  occur  in  the  third  of  formulas  (1).     Hence 

^  =  1  =  0.7778,  whence  A  =  37^87,  and  hence  B  =  90°  -  37^87 
0      9 

52°.13.    To  find  c  we  have 


tan  A 


a=7 


CSC  A  =  -,  whence  c  =  a  esc  A 

^  =7x1.6290  =  11.403, 


Check.    Find  b  from  B  and  c.    Since 
sin  B  =  b/c,  we  have 

6  =  c  sin  B  =  11.403  x  0.7894  =  9.0015, 
which   agrees  reasonably  well   with   the 
given  value  of  b. 

It  is  to  be  noted  that  the  angles  and  fines  in  these  examples 
and  in  the  exercises  following  have  not  been  measured  with 
the  same  degree  of  precision.  The  angles  are  given  to  four 
figures  in  order  to  afford  practice  in  interpolation. 

Problems  involving  isosceles  triangles  and  regular  polygons 
may  be  solved  by  means  of  right  triangles,  for  such  figures  may 
be  divided  into  congruent  right  triangles. 


TRIGONOMETRIC  FUNCTIONS 


183 


Fig.  102. 


65.  Applications.  An  instrument  known  as  a  transit  enables 
surveyors  to  measure  angles  in  vertical  and  horizontal  planes. 

If  A  and  B  are  points  not  in  the  same  horizontal  plane,  and 
if  AC  and  BD  are  horizontal  lines  in  the  vertical  plane  through 
A  and  B,  then  Z  BAG  is  called  the  angle  of  depression  of  B  at 
A,  and  ZABD  is  called  the  angle  of  elevation  of  A  at  B.  If  the 
eyes  are  at  A,  looking  horizontally 
over  B,  the  angle  of  depression  of  B 
is  the  angle  through  which  the  eyes 
must  be  lowered  to  see  B.  While 
if  the  eyes  are  at  B,  looking  hori- 
zontally below  A ,  the  angle  of  eleva- 
tion of  A  is  the  angle  through  which 
the  eyes  must  be  raised  to  see  A. 

If  A  A'  and  BB'  are  the  vertical  lines  through  two  points  A 
and  B,  meeting  the  horizontal  plane  through  a  third  point  C 

at  A'  and  B'  respectively,  then 
/.A'CB'  is  called  the  horizontal 
angle  between  A  and  B  at  C.  If  we 
think  of  A  A '  and  BB'  as  two  trees, 
of  different  heights  usually,  for  an 
observer  at  C,  the  horizontal  angle 
between  the  tree  tops  A  and  B  is 
the  angle  through  which  one  must 
turn,  if  one  faces  first  toward  the  tree  A  A'  and  then  turns  to 
face  the  tree  BB'. 

The  hearing  of  a  line  is  the  angle  the  line  makes  with  some 
fundamental  line  of  the  figure  which  is  called  the  base  line. 
For  example,  if  the  base  fine  is  north  and  south,  the  bearing 
of  a  fine  running  northeast  is  45°  east  of  north. 


Fig.  103. 


EXERCISES 

1.  In  any  right  triangle,  if  c  and  A  are  given,  show  that  a  =  c  sin  A 
and  6  =  c  cos  A.  Express  these  equations  in  words.  If  a  and  A  are  given, 
find  b  and  c.     If  h  and  A  are  given,  find  a  and  c. 


184  ELEMENTARY  FUNCTIONS 

2.  Solve  and  check  the  following  right  triangles: 

(a)  A  =  22°.13,  6  =  5.       (h)  B  =  40°.28,  c  =  4.         (c)  6  =  17,  c  =  30. 
(d)  a  =  10,       5  =  19.      (e)  A  =  66^47,  =  20.         (f)  a  =  30,  c  =  37. 

3.  To  find  the  width  of  a  river,  two  points,  A  and  C,  are  taken  on  one 
bank  100  feet  apart.  If  5  is  the  point  on  the  other  bank  directly  opposite  1 
C,  and  if  ZCAB  equals  72°.  16,  how  wide  is  the  river? 

4.  From  the  top  of  a  lighthouse  35  feet  high,  the  angle  of  depression  i 
of  a  ship  is  11°.38.     How  far  is  the  ship  from  the  lighthouse? 

5.  What  is  the  angle  of  elevation  of  the  sun  if  a  pole  49  feet  high  casts 
a  shadow  of  11  feet? 

6.  An  iron  wedge  for  splitting  rails  is  to  have  a  base  two  inches  wide 
and  a  vertex  angle  of  15°.     How  long  will  each  side  be? 

7.  A  rustic  summer  house,  or  shelter,  is  to  be  built  with  an  octagonal 
floor,  6  feet  on  a  side.  Determine  the  amount  of  flooring  necessary,  mak- 
ing an  allowance  of  25  %  on  account  of  the  "  tongue  and  groove  "  and  for 
waste. 

8.  Solve  Exercise  7  if  the  floor  is  to  be  pentagonal.  ^Ive  Exercise  7 
if  the  floor  is  to  be  a  regular  seven-sided  polygon.  Can  this  Exercise  be 
solved  by  plane  geometry? 

9.  If  the  radius  of  a  regular  polygon  of  n  sides  is  r,  find  (a)  the  perimeter 
in  terms  of  n  and  r;   (b)  the  area. 

10.  If  a  side  of  a  regular  polygon  is  a,  express  the  area  as  a  function  of 
a,  assuming  n  to  be  constant. 

11.  What  is  the  angle  of  inclination  of  an  ordinary  gable  roof,  if  its 
pitch  (the  ratio  of  its  height  to  its  entire  width)  is  f  ?  ^? 

12.  A  point  P  moves  with  uniform  speed  around  a  circle  of  radius  one 
foot,  making  a  complete  revolution  every  36  seconds.    The  projection  M 

of  P  on  a  diameter  AOB,  moves  along  the 
diameter.  Find  OM  for  the  values  50°,  60°, 
70°,  if  ^  =  ZAOP.  Find  the  average  velocity 
of  M  as  ^  increases  from  50°  to  60°  and  from 
60''  to  70°.  Find  the  average  velocity  of  M  as 
6  increases  from  59°  to  60°  and  from  60°  to 
61°.  From  59°.9  to  60°  and  from  60°  to  60°.l. 
Approximately  what  is  the  velocity  of  M  at 
the  instant  when  6  =  60°? 

13.  Two  life  saving  stations  are  10  miles 
apart  on  a  beach  running  22°.5,  east  of  north. 
A  lightship  anchored  off  the  beach  lies  33°.75  north  of  east  from  one 
station,  and  11°.25  east  of  south  from  the  other.  How  long  would  it  take 
a  boat  to  run  from  one  station  to  the  other  if  its  speed  is  10  miles  per 
hour,  and  if  it  passes  outside  the  lightship?  How  long  would  it  take  the 
boat  to  go  from  the  ship  to  the  nearest  point  on  the  beach? 

14.  (a)   Two  sides  of  a  triangle,  not  a  right  triangle,  are  12  and  20» 


TRIGONOMETRIC  FUNCTIONS 


185 


and  the  included  angle  is  28°.48.     Find  the  altitude  on  the  latter  side  and 
the  area. 

(b)  Find  the  area  of  any  triangle  ABC  in  terms  of  the  sides  b  and  c  and 
the  included  angle  A. 

15.  Find  the  area  of  a  parallelogram  in  terms  of  two  adjacent  sides  and 
the  included  angle. 

16.  Two  ships  are  on  a  line  w^ith  a  lighthouse,  which  is  40  feet  high. 
At  the  top  of  the  lighthouse,  the  angles  of  depression  of  the  ships  are 
respectively  4°.26  and  6°.31.     How  far  apart  are  the  ships? 

17.  A  regular  octagonal  tower  is  4  feet  on  each  side.  The  roof  is  10 
feet  high.     How  long  should  the  rafters  be? 

18.  A  bridge  is  to  be  built  across  a  ravine  between  two  points  A  and  B 
on  the  same  level.  Two  stations  C  and  D  are  chosen  in  the  ravine  which 
lie  in  the  vertical  plane  through  A  and  B.  At  A,  the  angle  of  depression 
of  C  is  42°.37,  and  AC  =  50  feet.  At  C  the  angle  of  depression  of  D  is 
4**.52,  and  CD  =  40  feet.  At  D  the  angle  of  elevation  of  B  is  37°.89,  and 
DB  =  60  feet.     Find  the  length  of  the  bridge. 

19.  Check  the  accuracy  of  the  measurements  in  Exercise  18,  by  finding 
the  difference  in  altitude  of  A  and  D,  and  also  of  B  and  D. 

20.  To  find  the  width  of  a  river,  a  tree  is  selected  on 
one  shore.  At  a  point  50  feet  from  the  tree,  the  angle  of 
elevation  of  the  top  is  48°.72.  At  the  point  on  the  other 
shore  directly  opposite  the  tree,  the  angle  of  elevation  of 
the  top  is  17**.39.      How  wide  is  the  river? 

21.  From  a  station  A  the  horizontal  angle  between  a 
mountain  top  C  and  a  second  station  B,  1000  feet  from  A 
and  at  the  same  altitude,  is  64''.37.  At  B  the  horizontal 
angle  between  the  mountain  top  and  A  is  90°,  and  the  angle 
of  elevation  of  the  mountain  top  is  37*^.24.     How  high  is  the  mountain? 

66.  Parallelogram  Law  —  Velocities,  Accelerations,  Forces. 

The  law  to  be  considered  in  this  section  is  illustrated  in 

Example  1.  The  current  in  a  river  flows  at  the 
rate  of  2  miles  an  hour.  A  man  rows  across  at  the 
rate  of  4  miles  an  hour,  keeping  his  boat  at  right 
angles  to  the  shore.  Show  that  the  boat  moves 
in.  a  straight  line.  Find  how  fast  it  moves,  and  in 
what  direction. 

Take  the  starting  point  for  the  origin  of  a  system 
of  coordinates,  and  let  the  x-axis  lie  along  the  bank 
of  a  river.  At  the  time  t,  the  boat  will  be  at  a 
point  P,  whose  coordinates  give  the  distance  the 
boat  has  been  carried  by  the  current,  x  =  2t,  and 
*the  distance  the  man  has  rowed  from  shore,  y  =  4^.    Eliminating  t,  we 


JllOOOft.S 

Fig.  105. 


AM 
Fig.  106. 


186  ELEMENTARY  FUNCTIONS 

obtain  y  =  2x,  which  is  true  for  all  values  of  t.  Hence  the  boat  moves  in 
a  straight  line  (Corollary  1,  page  57). 

Let  C  denote  the  position  of  the  boat  one  hour  after  starting,  and  let 
OA  and  OB  be  the  coordinates  of  C. 

The  line  OA  represents  the  velocity  of  the  water  with  reference  to  the 
earth,  the  direction  of  the  line  being  that  in  which  the  water  flows,  and  the 
length  of  the  hne  being  the  niunber  of  miles  per  hour  the  water  moves. 
Similarly,  OB  represents,  in  direction  and  magnitude,  the  velocity  of  the 
boat  vrith  reference  to  the  water.  (If  the  water  were  at  rest,  OB  would  repre- 
sent the  actual  motion  of  the  boat,  in  one  hour,  with  reference  to  the 
earth.)  With  reference  to  the  earth,  the  boat  moves  along  the  line  OC,  and 
in  one  hour  moves  from  0  to  C  Hence  OC  represents,  in  direction  and 
magnitude,  the  actual,  or  resultant,  velocity  of  the  boat. 

The  resultant  velocity  is  readily  computed.    Its  magnitude  is 


OC  =  VOA^  +  AC^  =  V22  +  42  =  4.47  miles  per  hour. 
Its  direction  may  be  given  by  ZAOC.    Since 

tan  AAOC  =  AC/OA  =  4/2  =  2,  we  have  ZAOC  =  63°.43. 

This  illustration  exemplifies  the  law  known  as  the 

Parallelogram  of  velocities.  If  a  body  is  subjected  to  two  differ- 
ent  velocities,  represented  in  direction  and  magnitude  by  two  lines 
OA  and  OB,  the  actual  or  resultant  velocity  i^s  represented  by  the 
diagonal  OC  of  the  parallelogram  determined  by  OA  and  OB. 

If  a  body  is  moving  through  the  air  in  any  way,  it  has  an  accel- 
eration of  32  feet  per  second  per  second  directed  toward  the 
center  of  the  earth  (see  Section  22,  page  63).  This  means  that 
its  vertical  velocity  is  increased  or  decreased  each  second  by  32  feet 
per  second,  according  as  it  is  falling  or  rising.  Its  horizontal  ve- 
locity is  uniform,  and  is  not  affected  by  the  action  of  gravity. 
No  account  is  taken  here  of  the  resistance  of  the  air,  which 
would  make  the  discussion  very  much  more  complicated. 

The  acceleration  due  to  gravity  may  be  represented  by  a 
vertical  line  running  downward  whose  length  is  32.  In  hke 
manner,  any  acceleration  may  be  represented,  in  direction  and 
magnitude,  by  a  line. 

A  force  may  also  be  represented,  in  direction  and  magnitude, 
by  a  line  OA,  0  representing  the  point  of  application  of  the  force. 

Accelerations  and  farces  may  also  be  combined  according  to 
the  parallelogram  law. 


TRIGONOMETRIC  FUNCTIONS 


187 


Finding  the  resultant  of  two  velocities  is  known  as  the  compo- 
sition of  velocities,  the  two  given  velocities  being  called  com- 
ponents of  the  resultant.  The  converse  problem  of  determining 
two  component  velocities  which  have  a  given  resultant  is  called 
the  resolution  of  velocities.  The  same  terms  are  also  used  with 
reference  to  accelerations  and  forces. 

If  the  components  are  at  right  angles,  the  parallelogram  is  a 
rectangle,  and  the  composition  or  resolution  may  be  effected 
by  solving  a  right  triangle.  If  the  parallelogram  is  not  a 
rectangle,  it  is  necessary  to  use  the  methods  to  be  developed  in 
Sections  71-73.  Problems  of  this  sort  will  be  found  in  the 
exercises  following  Section  73. 

Example  2.  A  ball  rolls  down  a  plane  whose  inclination  is  30°.  Re- 
solve the  acceleration  due  to  gravity  into  two  components  parallel  and 
perpendicular  to  the  plane. 

Let  OC  =  32  represent  the  acceleration  due  to  gravity.  Through  0 
and  C  draw  lines  parallel  and  perpendicular  to  the  plane  DE,  forming  the 
rectangle  OABC.  Then  OA  and  OB  represent  the  required  components, 
for  the  resultant  of  OA  and 
OB  is  OC  (parallelogram  law). 

In  the  triangle  OAC,  OC 
=  32,  and  ZACO  =  ZEDF 
=  30°  (why?).  A 

Hence         OA/32  =  sin  30* 
and  AC/32  =  cos  30^ 

Then  OA  =  32  x  |  =  16 

and     05  =  AC  =  32  X  W3 
=  16\/3. 

As  there  is  no  motion  in  a  direction  perpendicular  to  the  plane,  the 
component  OB  is  neutralized  by  the  plane.  The  component  parallel  to 
the  plane,  OA  =  16  feet  per  second  per  second,  gives  approximately 
the  effective  acceleration  with  which  the  ball  rolls  down  the  plane.  If 
the  ball  starts  to  roll  from  rest,  how  fast  will  it  be  moving  at  the  end  of 
one  second?  At  the  end  of  two  seconds?  At  the  end  of  four  seconds? 
If  a  ball  is  rolled  up  the  plane,  how  will  its  velocity  be  affected  during 
any  second?  If  it  is  started  up  with  a  velocity  of  50  feet  per  second,  how 
long  will  it  roll  up  the  plane? 

67.  Conditions  of  Equilibrium  of  a  Particle.     In  measuring 
,  a  force  we  shall  use  the  pound  as  the  unit. 


Fig.  107. 


I 


188 


ELEMENTARY  FUNCTIONS 


If  a  number  of  forces  act  on  a  particle,  which  we  take  as  the 
origin  of  a  system  of  coordinates,  each  of  the  forces  may  be 
resolved  into  two  components,  one  acting  along  each  axis. 
If  the  particle  is  in  equihbrium,  there  is  no  motion  in  the  direc- 
tion of  the  X-axis,  and  hence  the  algebraic  sum  of  the  com- 
ponents along  that  axis  must  be  zero.  Similarly,  the  sum  of 
the  components  along  the  y-Sixis  must  be  zero.  And  conversely, 
if  each  of  these  sums  is  zero  the  particle  must  be  in  equilibrium. 
Hence  we  have  the 

Theorem.  A  particle  is  in  equilibrium  under  the  action  of 
any  number  of  forces  if  and  only  if  the  sum  of  the  components  in 
each  of  two  perpendicular  directions  is  zero. 

In  applying  this  theorem,  first  determine  all  the  forces 
acting  on  the  particle,  and  then  choose  the  perpendicular  direc- 
tions. If  two  of  the  forces  are  at  right  angles  to  each  other, 
choose  these  directions. 

Example  1.  A  particle  weighing  4  ounces  is  supported  on  a  smooth 
plane  whose  inclination  is  30°  by  a  cord  parallel  to  the  plane.  Find  the 
tension  of  the  cord  and  the  pressure  on  the  plane. 

The  forces  acting  on  the  particle 
are 

(1)  Its  weight,    TF  =  4,  acting 
vertically. 

(2)  The  tension  of  the  cord,  jT, 
acting  parallel  to  the  plane. 

^(3)  The  resistance  of  the  plane, 
R,  acting  perpendicular  to  the 
plane. 

Since  R  and  T  act  at  right 
angles,  we  proceed  to  resolve  all 
the  forces  into  components  paral- 
lel and  perpendicular  to  the  plane.  Choosing  these  directions  facilitates 
the  work,  because  W  is  then  the  only  force  whose  components  must  be 
determined.  By  the  parallelogram  law  and  the  solution  of  a  right 
triangle,  the  component  of  W  parallel  to  the  plane  is  found  to  be  4  sin  30**, 
and  that  perpendicular  to  the  plane  is  4  cos  30**,  both  acting  downward. 
Then  by  the  theorem,  taking  the  directions  of  R  and  T  a»  positive, 
we  have 

r  -  4  sin  30**  =  0,        and        R  -4  cos  30**  =  0, 
whence  T  =  2  ounces  and       R  =  2\/3  =  3.4  ounces. 


Fig.  108. 


TRIGONOMETRIC  FUNCTIONS  189 

In  pointing  out  the  part  played  by  mathematics  in  the 
illustration  of  the  scientific  method  in  Example  1,  page  79 
(see  bottom  page  80),  it  was  stated  that  a  more  satisfactorj^ 
verification  of  the  law  obtained  would  be  indicated  later,  by 
deducing  from  the  law  some  fact  that  may  be  verified  by  an 
experiment  of  a  different  nature.  •  This  deduction  will  be  macle 
in  Example  2  below. 

Friction  between  an  object  and  a  plane  acts  parallel  to  the 
plane,  and  in  the  direction  opposite  to  that  in  which  the  object 
moves  or  tends  to  move.  When  motion  is  about  to  take  place, 
the  force  of  friction  is  equal  to  the  coefficient  of  friction  for 
the  surfaces  in  contact  (see  page  81)  multiplied  by  the  pressure 
of  the  object  on  the  plane  in  the  direction  perpendicular  to  the 
plane.     It  is  this  law  that  we  wish  to  verify. 

Example  2.  A  block  of  wood  weighing  20  grams  rests  on  a  horizontal 
board.  If  the  coefficient  of  friction  is  0.29,  the  result  obtained  in  Example 
1,  page  79,  at  what  angle  may  the 
board  be  tipped  before  the  block 
will  be  on  the  point  of  sliding? 

The  forces  acting  on  the  bloc 
are: 

(1)  The  resistance  of  the  plane,  fi, 

(2)  The  friction,  F,  and 

(3)  The  weight  W  =  20. 

It  is  known  that 

Fig.  109. 
F  =  0.29  R,  (1) 

since  the  pressure  on  the  plane  is  numerically  equal  to  the  resistance  of 
the  plane. 

Resolving  W  into  components  parallel  and  perpendicular  to  the  plane, 
as  in  Example  1  above,  we  get 

F  =  20  sin  e,        R  =  20  cos  d,  (2) 

where  6  is  the  angle  at  which  the  block  is  on  the  point  of  sliding. 
Substituting  in  (1)  the  values  of  F  and  R  given  by  (2),  we  get, 

20  sin  d  =  0.29  X  20  cos  d. 

Dividing  both  sides  by  20  cos  6,  we  obtain 

?i5-^  =  o.29. 
cos  u 


190  ELEMENTARY  FUNCTIONS 

By  equation  (2)  on  page  167,  n  =  tan  6. 

Hence  tan  6  =  0.29, 

whence  6  =  16*'.2. 

This  result  may  be  readily  tested  by  experiment.  If  it  is 
found  that  when  one  end  of  the  board  is  gradually  raised  the 
block  begins  to  slide  when  the  incUnation  is  a  httle  over  16°, 
then  we  have  a  verification  of  the  correctness  of  the  deduction 
by  which  6  was  found,  and  also  a  verification  of  the  law  (1)  on 
which  the  deduction  was  based.  That  is,  we  have  a  verifica- 
tion of  the  law  obtained  in  Example  1,  page  79,  which  was 
re-stated  on  page  81. 

EXERCISES 

1.  A  boy  kicks  a  football  so  that  it  would  roll  across  a  street  at  the  rate 
of  15  feet  per  second,  and  simultaneously  a  second  boy  kicks  it  so  that  it 
would  roll  along  the  street  with  a  velocity  of  12  feet  per  second.  Find 
the  actual  velocity  of  the  ball  (magnitude  and  direction). 

2.  A  man  walks  across  a  canal  boat.  If  his  velocity  with  reference  to 
the  earth  is  7  feet  per  second  in  a  direction  inclined  at  30**  to  the  bank  of 
the  canal,  find  the  velocity  of  the  boat  and  that  at  which  he  walks. 

3.  A  ball  is  thrown  into  the  air  at  an  angle  of  40*  with  a  velocity  of  30 
feet  per  second.  Find  the  horizontal  and  vertical  components  of  the 
velocity. 

4.  If  a  ball  is  placed  on  a  plane  inclined  at  20°,  find  the  acceleration 
with  which  it  rolls  down  the  plane,  and  how  fast  it  will  be  moving  at  the 
end  of  3  seconds. 

5.  A  ball  is  rolled  up  a  plane  inclined  at  10**  with  an  initial  velocity  of 
20  feet  per  second.     How  long  will  it  roll  up  the  plane? 

6.  What  is  the  inclination  of  a  sidewalk  if  a  ball  rolls  down  it  with  an 
acceleration  of  8  feet  per  second  per  second? 

7.  A  cake  of  ice  weighing  200  pounds  is  held  on  an  ice  slide  at  an  ice 
house  by  a  rope  parallel  to  the  slide.  If  the  inclination  of  the  shde  is  45", 
find  the  pull  on  the  rope  and  the  pressure  on  the  slide. 

8.  An  automobile  weighing  2500  pounds  stands  on  a  pavement  in- 
clined at  12**.    What  force  do  the  brakes  exert? 

9.  Will  a  box  slide  down  a  board  inclined  at  15**  if  the  friction  is  0.3  of 
the  pressure  of  the  box  on  the  plane? 

10.  A  bar  of  iron  weighing  5  pounds  rests  on  a  rough  board.  As  one 
end  of  the  ^  .ard  is  gradually  raised,  it  is  found  that  the  bar  is  just  ready 
to  slide  when  the  inclination  is  20**.  Find  the  force  of  friction  and  the 
coefficient  of  friction  (Definition,  page  81). 


TRIGONOMETRIC   FUNCTIONS  191 

11.  A  boy  and  his  sled  weigh  50  pounds.  What  force  is  necessary  to 
hold  them  on  an  icy  sidewalk  whose  inclination  is  10°  by  means  of  a  rope 
inclined  at  30**  to  the  walk? 

12.  (a)  A  rifle  is  fired  at  an  angle  of  20°  to  the  horizon.  If  the  muzzle 
velocity  of  the  ball  is  2000  feet  per  second,  find  the  horizontal  and  vertical 
components  of  the  velocity  at  the  muzzle. 

(b)  How  long  will  a  ball  rise  if  it  is  thrown  vertically  upward  with  a 
velocity  of  32  feet  per  second?  of  80  feet  per  second?  How  long  will  the 
rifle  ball  in  (a)  rise?    How  far  will  it  move  horizontally  in  this  time? 

13.  A  rifle  with  muzzle  velocity  of  1600  feet  per  second  is  fired  at  an 
inclination  of  30°.  Find  the  horizontal  and  vertical  components  of  the 
velocity  at  the  muzzle.  How  long  will  it  rise?  When  will  it  hit  the  ground 
(see  Exercise  13,  page  104)?  How  far  from  the  point  where  it  is  fired  will 
it  hit? 

14.  A  rope  is  tied  to  a  heavy  weight  lying  on  the  ground.  A  boy  pulls 
on  the  rope  with  a  force  of  50  pounds  in  such  a  way  that  the  rope  is  in- 
clined at  25°  to  the  vertical.  Find  the  horizontal  and  vertical  compo- 
nents of  his  pull.  What  is  the  force  with  which  he  tends  to  lift  the  weight? 
To  drag  it?  If  the  weight  weighs  100  pounds,  what  is  the  pressure  of  the 
weight  on  the  ground  when  the  boy  pulls? 

16.  A  body  weighing  25  pounds  rests  on  a  rough  plane  inclined  at  5°. 
Find  the  components  of  the  weight  parallel  and  perpendicular  to  the  plane. 
What  is  the  pressure  of  the  body  on  the  plane?  What  force  is  tending  to 
make  the  body  move  down  the  plane?  Why  does  it  not  move?  (The 
weight  of  a  body  is  the  force  with  which  the  earth  attracts  it.) 

16.  Three  bales  of  cotton,  a  total  weight  of  1400  pounds,  are  raised  from 
the  hold  of  a  ship  by  a  derrick  until  they  are  just  above  the  deck,  when  they 
are  pulled  one  side  by  a  horizontal;  rope.  Find  the  pull  of  the  rope  and 
of  the  cable  on  the  derrick  when  the  cable  is  inclined  at  12°  to  the 
vertical. 

17.  A  stone  weighing  300  pounds  is  raised  to  the  top  of  a  building  by 
means  of  a  derrick  on  top  of  the  building,  while  a  man  on  the  ground  keeps 
it  away  from  the  side  of  the  building  by  means  of  a  rope  tied  to  the  stone. 
Find  the  pull  on  the  cable  of  the  derrick  and  on  the  rope  if  the  cable  is 
inclined  to  the  vertical  at  5°  and  the  rope  at  60°. 

18.  Two  cords  are  tied  to  a  weight  of  10  pounds.  The  weight  is  held 
by  two  boys  who  pull  on  the  cords.  If  one  cord  is  inclined  at  20°  to  the 
vertical  and  the  other  at  14°,  find  the  force  with  which  each  boy  pulls. 

19.  Two  narrow  boards  are  2  feet  long.  A  boy  places  their  upper  ends 
together  and  their  lower  ends  on  the  ground  1  foot  apart,  and  balances  a 
weight  of  5  pounds  on  top  of  them.     Find  the  pressure  along  each  board. 

20.  A  body  weighing  5(X)  pounds  is  suspended  from  the  center  of  a 
horizontal  beam.  The  beam  rests  on  two  V-shaped  supports,  inverted, 
which  are  made  of  pieces  of  "2x4"  each  4  feet  long.    Find  the  thrust 


192 


ELEMENTARY  FUNCTIONS 


or  pressure  along  each  2x4,  assuming  that  the  effect  is  the  same  as  if 
half  of  the  weight  were  placed  directly  over  each  support. 

21.  Just  above  the  door  on  the  second  story  of  a  barn,  a  beam  projects 
horizontally  for  3  feet.  Objects  are  raised  to  the  second  story  by  a  rope 
which  passes  over  a  pulley  at  the  end  of  the  beam  and  enters  the  bam 
over  a  pulley  in  the  wall  at  a  point  4  feet  directly  above  the  beam.  Find 
the  thrust  along  the  beam  when  an  object  weighing  60  pounds  is  suspended 
by  the  rope. 

68.  Functions  of  n90°  =t  6,  In  Section  62  we  saw  how  to 
express  the  functions  of  the  complement  of  6,  90°  —  6,  in  terms 
of  functions  of  6.  Let  us  now  consider  the  functions  of  180°  -  6, 
the  supplement  of  6,  employing  the  same  method. 

Construct  the  angles  6  and  180°  -  6  with  their  initial  Hnes 
coinciding  with  the  x-axis,  and  on  their  terminal  lines  take 
P(x,  y)  and  P'(x\  y')  so  that  r  =  r'.  Then  P  and  P'  are  sym- 
metrical with  respect  to  the  2/-axis  (why?),  and  hence  x'  =  -  Xy 

and  y'  =  y.    Then 


>s. 

P' 

y 

' 

P^ 

y' 

\ 

X 

V    180 

y 

3 

1' 

x' 

0 

X 

iV 

I    X 

Fig.  110. 


sin  (180°  -6)  =% 

=  sin  6. 
cos(180°-^)=|-i  = 
=  -  cos  6, 


y 

r 

(1) 

X 

r 

(2) 


Dividing  (1)  by  (2),  and  applying  (2),  page  167,  we  obtain 

tan  (180°  -  ^)  =  -  tan  ^.  (3) 

Similar  formulas  may  be  obtained  for  the  other  functions 
from  these  formulas  by  means  of  the  reciprocal  relations. 

Inspection  of  these  re- 
sults shows  that 

The  numerical  values  of 
the  trigonometric  functions 
of  an  obtuse  angle  are  equal 
respectively  to  the  functions 
of  the  supplementary  angU. 

In  the  figure  above,  6  Fio.  ill. 


M 

X 

'A 

^. 

x' 

M' 

A 

-e 

X 

y 

/, 

r/ 

SV 

N 

y 

J 

p 

p 

k 

TRIGONOMETRIC   FUNCTIONS 


193 


ft 


was  taken  acute.  But  the  proof  applies  without  change  to  the 
second  figure,  in  which  d  is  in  the  third  quadrant.  It  would 
also  apply  to  properly  constructed  figures  in  which  6  lies  in 
the  second  or  fourth  quadrants,  so  that  these  formulas  hold 
for  all  positive  values  of  6  less  than  360°.  The  proof  may  be 
extended  to  include  all  values  of  d  by  using  the  periodicity  of 
the  functions. 
Any  function  of  one  of  the  angles 

-  d,  90°  =t  e,  180°  ^  d,  270°  =t  6,  360°  -  6 

may  be  expressed  in  terms  of  a  function  of  6.  Any  one  of 
these  forty-eight  formulas  may  be  written  down  by  the  fol- 
lowing rule: 

If  any  one  of  the  trigonometric  functions  he  denoted  by  f(6), 
and  its  co-function  (page  177)  by  co-f{6),  then 

=*=  /(^)  if  ri  is  zero  or  even,  or 
,  ±  co-fid)  if  n  is  odd. 


/(n90°  ^  6) 


? 

y 

p 

2/1 

X 

\<1 

ry^ 

y 

M^ 

«1 

^!w     X 

M 

x^ 

>\^         ^3 

X 

2/2 

^ 

^Jy^ 

^x:3 

2/8 

i 

2 

1 

3 

Fig.  112. 


The  sign  on  the  right 
is  to  agree  with  that  of 
/(n90°  =t  d)  when  d  is 
acide. 

A  comprehensive 
survey  of  these  rela- 
tions   is   afforded    by 
the   figures  following. 
To  construct  the  first 
figure,    construct    the 
angles   6,   180°  -  6,   180°  +  6,  and   360°  -  6,  OX  being  the 
initial  line  of  each.     On  their  terminal  Unes  take,  respectively, 
f*(^,  2/),  P\ixi,  2/i),  P2{x2, 2/2),  and  Pz{xz,  ys),  so  that 
r  =  n  =  r2  =  rz. 

Then  Pi,  P2,  and  P3  are  symmetrical  to  P  with  respect  to 
the  y-axis,  the  origin,  and  the  x-axis  respectively  (why?). 
Hence  the  numerical  values  of  x,  Xi,  X2,  and  X3  are  equal,  and 
80  also  are  those  of  ?/,  2/1,  2/2,  and  ys.  Hence  the  numerical  value 
of  any  function  of  one  of  the  angles  180°  -  6, 180°  +  d,  or  360°  -  6 


194 


ELEMENTARY  FUNCTIONS 


is  equal  to  that  of  the  same  function  of  6.  Whether  the  signs 
agree  or  not  will  depend  on  the  angle  and  the  function  in 
question.     For  example: 

Since  2/2  =  2/3  =  -?/,  we  have  ^  =  ^^  =  _  ^ 

whence     sin  (180°  +  ^)  =  sin  (360°  -  6)  =  -  sin  B.  (4) 

And  since  2/2  =  -  y,  and  X2  =  -  a;,  so  that  2/2/^2  =  y  jx,  we  have 

tan  (180°  +  6)  =  tan  6.  (5) 

To  construct  the  second  figure,  lay  off  the  angles  6,  90°  -  6, 

90°  +  B,  270°  -  B,  and  270°  +  B,  with  OX  for  the  initial  line. 

On  their  terminal  lines  take 
P{x,  y),  Pi(xi,  2/1),  ^2(^:2,  2/2), 
Psixs,  2/3)  and  P4{x4,  2/4),  so  that 
r  =  n  =  r2=  rs  =  r4. 
Since  Pi  is  symmetrical  to  P 
with  respect  to  the  bisector  of  the 
first  quadrant  (why?), 

Xi  =  y  and  2/1  =  x. 

P2,  P3,  and  P4  are  symmetrical 
to  Pi  with  respect  to  the  2/-axis, 
the  origin,  and  the  a>-axis  respec- 
tively. Hence  the  numerical  values  of  Xi,  x^,  X3,  and  Xi  are 
equal  to  y,  and  those  of  2/1,  2/2, 2/3,  and  2/4  are  equal  to  x.  In- 
spection of  the  definitions  on  page  166  shows  that  if  x  and  y 
are  interchanged,  each  function  is  replaced  by  its  co-function. 
Hence  the  numerical  values  of  any  function  of  one  of  the  angles 
90°  =i=  B  or  270°  =*=  B  is  equal  to  that  of  the  co-function  of  B, 
For  example,  since  0:2  =  0:3  =  -  xi  =•  -  2/, 


2/2 

\ 

//Xr 

P 

y 

M2 

*2    \ 

m-^ 

Ml 

Xi  y 

\         «4 

xj^x 

2/3 

^3/ 

\ 

1 

Fig.  113. 


we  have 


y 


Ti      r$  r 

whence      cos  (90°  +  ^)  =  cos  (270°  -  ^)  =  -  sin  ^.  (6) 

In  this  survey  we  have  taken  B  acute,  so  that  all  the  func- 
tions of  B  are  positive.  Hence  fin90°  ^  B)  will  equal  ^  fiB), 
or  =*=  co-fiB),  according  as  /(n90°  =*=  B)  is  positive  or  negative, 
which  can  be  determined  readily  in  a  given  case. 


TRIGONOMETRIC   FUNCTIONS 


195 


To  illustrate  the  use  of  the  rule,  express  cos  (270°  —  6)  in 
terms  of  a  function  of  6,  Since  270°  =  3-90°,  and  3  is  odd,  we 
will  obtain  the  co-function,  ±  sin  6.  If  6  is  acute,  270°  -  6 
is  in  the  third  quadrant,  in  which  the  cosine  is  negative  (see 
graph).    Hence         cos  (270°  -  6)  =  -  sin  6.  (7) 

Writing  down  a  formula  by  the  rule  does  not  constitute  a 
proof.  The  proof  of  (7),  for  example,  which  is  identical  with 
part  of  (6),  is  given  above. 

The  graphical  significance  of  any  one  of  the  formulas  is  easily 
found.  Consider,  for  example,  formulas  (1)  and  (4),  and 
construct  the  graph  of  sin  6.  Let  OA  =  6,  and  construct 
OB  =  180°  -  d,OC  =  180°  +  e,OD  =  360°  -  6.  Then  by  (1), 
the  ordinates  at  A  and  B  are  equal,  and  by  (4),  the  ordinates  at 
C  and  D  are  equal  nu- 
merically to  that  at  Af 
but  are  opposite  in 
sign. 

If  6  increases  from 
0°  to  90°,  it  follows 
that  the  graph  consists 
of  four  congruent  parts. 

Periodicity  of  tan  6.  From  (5)  it  follows  at  once  that  the 
period  of  tan  6  is  180°,  or  tt. 

The  following  formulas  are  stated  for  purposes  of  reference: 

sin  (180°  +  ^)  =  -  sin  d.  (8) 

cos  (180° +  <9)  =  -cos^.  (9) 

tan  (180°  +  ^)  =      tan  d.  (10) 

sin  (360°  -  d)  =  -  sin  6.  (11) 

cos  (360°  -  ^)  =      cos  d.  (12) 

tan  (360°  -  6)  =  -  tan  d.  (13) 

If  360°  are  subtracted  from  the  angle  on  the  left  in  (11),  (12), 
and  (13)  we  obtain 

sin  (-  ^)  =  -  sin   d,  ^ 

cos  (-6)  =      cos  6,  [  (14) 

tan  (-  0)  =  -  tan  d,  J 

the  relations  proving  the  synometry  of  the  graphs. 


Fig.  114. 


196  ELEMENTARY  FUNCTIONS 

69.  Application  to  the  Use  of  Tables.  A  positive  angle  less 
than  360°  may  be  put  in  the  form 

180°  -  e,        180°  +  0,        or        360°  -  6,         (1) 

where  6  is  acute,  according  as  the  angle  Hes  in  the  second, 
third  or  fourth  quadrant.     Hence  the  functions  of  such  an 
angle  may  be  expressed  in  terms  of  the  functions  of  an  acute 
angle  6,  and  the  latter  may  be  found  from  the  tables. 
For  example: 

sin  150°  =  sin  (180°  -  30°)  =  sin  30°  =  i. 

cos  213°  =  cos  (180°  +  33°)  =  -  cos  33°  =  -  0.8387. 

tan  312°  =  tan  (360°  -  48°)  =  -  tan  48°  =  -  1.1106. 

The  functions  of  a  positive  angle  greater  than  360°  may  be 
found  by  using  the  periodicity  of  the  function,  and  then  pro- 
ceeding as  above;  for  example, 

sin  985°  =  sm  (2-360°  +  265°)  =  sin  265°  =  sin  (180°  +  85°) 
=  -  sin  85°  =  -  0.9962. 

The  functions  of  a  negative  angle  are  found  by  first  using 
either  the  relations 

sin  {-  B)  =  -  sin  6,  cos  (-  6)  =  cos  6,  tan  (-  0)  =  -  tan  0, 
etc.,  or  the  periodicity.    Thus 

tan    (-  225°)  =  -  tan    225°  =  -  tan    (180°  +  45^) 
=  -  tan  45°  =  -  1, 
or  tan  (-  225°)  =  tan  (360°  -  225°)  =  tan  135^ 

=  tan  (180°  -  45°)  =  -  tan  45°  =  -  1. 

The  fimctions  of  a  positive  acute  angle  less  than  360°  may 
also  be  found  by  putting  the  angle  in  one  of  the  forms 

90°  +  d,        270°  -6,        or        270°  +  6,  (2) 

where  6  is  acute.  The  form  (1)  is  somewhat  less  confusing 
because  the  application  of  the  rule  in  Section  68  does  not  in- 
volve a  change  to  the  co-function. 

The  examples  following  show  how  to  find  all  the  angles  for 
which  a  given  function  has  a  given  value.     The  solutions 


TRIGONOMETRIC   FUNCTIONS  197 

depend  upon  the  fact  that  the  numerical  value  of  a  function  is 
the  same  for  the  three  angles  (1)  as  for  the  acute  angle  6. 

Example  1.     Find  all  values  of  B  for  which  sin  B  =  0.4332. 

First  find  the  positive  values  less  than  360°.  From  the  tables,  one  value 
is  ^  =  25°.67.  The  graph  of  sin  B  shows  that  a  line  parallel  to  the  0-axis 
and  0.4332  unit  above  it  cuts  the  graph  in  two  points,  one  in  the  first 
quadrant,  corresponding  to  the  value  B  =  25°.67  found  from  the  tables, 
and  one  in  the  second  quadrant.  Formula  (1)  Section  68,  shows  that  the 
second  value  is  0  =  180°  -  25°.67  =  154°.33. 

All  values  of  B  for  which  sin  ^  =  0.4332  are  given  by 

B  =  25°.67  +  n360°  and  B  -  154°.33  +  n360°. 

Example  2.     Find  ^  if  cos  ^  =  -  0.5. 

If  we  neglect  the  negative  sign  and  seek  an  acute  angle  whose  cosine  is 
0.5,  we  know  it  to  be  60°  (table,  page  161).  The  graph  of  cos  B  shows  that 
a  line  parallel  to  the  B-axis  and  0.5  unit  below  it  cuts  the  graph  in  two 
points,  one  in  the  second  and  one  in  the  third  quadrant.  Then  formulas 
(2)  page  192,  and  (9),  page  l95,  show  that  the  values  of  B  correspond- 
ing to  these  points  are  B  =  180°  -  60°  =  120°,  and  B  =  180°  +  60°  =  240°. 
All  values  of  B  are  then,  by  the  periodicity  of  cos  B, 

B  =  120°  +  n360°  and  B  =  240°  +  n360°. 

If  we  notice  that  one  of  the  angles  of  the  second  set  is,  for  n  =  -  1, 
-  120°,  which  may  also  be  obtained  from  B  =  120°  by  the  relation 
cos  {-  B)  =  cos  B,  all  the  angles  may  be  expressed  by  the  single  equation 

B=  ^  120°  +  n360°. 

Example  3.     Find  all  the  values  of  B  if  tan  B  =  -  V3. 

Neglecting  the  negative  sign,  we  recognize  that  one  value  of  B  is  60° 
(table,  page  161).  A  line  below  the  rc-axis  cuts  the  graph  of  tan  B  in  two 
points,  one  in  the  second  quadrant,  and  one  in  the  fourth.  Hence,  by 
formulas  (3)  page  192,  and  (13),  page  195,  the  required  values  of  B  less 
than  360°  are  180°  -  60°  =  120°  and  360°  -  60°  =  300°.  Then  all  the  re- 
quired values  are 

B  =  120°  +  n360°        and         B  =  300°  +  n360°. 

If  we  use  the  fact  that  the  period  of  tan  B  is  180°,  noticing  that 
300°  =  120°  +  180°,  all  these  angles  may  be  expressed  by  the  single 
equation 

B  =  120°  +  nl80°. 

EXERCISES 

1.  Prove  formulas  (8)  -  (13),  Section  68,  for  B  acute. 

2.  State,  prove,  and  give  the  graphical  significance  of  the  formula  for 


198  ELEMENTARY  FUNCTIONS 

(a)  cos  (90°  +  0).        (b)  cot  (-  0).  (c)  sec  (180"  -  0), 

(d)  sin  (270°  -  0).       (e)  cot  (360°  -  0).  (f)   cec  (90°  -  0). 
(g)  cot  (180°  +  0).      (h)  sec  (-  0).                   (i)   cos  (270°  +  0). 

3.  Prove  the  six  formulas  for  the  functions  of  the  angle: 

(a)  -  0.  (b)  180°  -  0.         (c)  180°  +  0.         (d)  360°  -  0. 

(e)  90°  -  0.  (f )   90°  +  0.  (g)  270°  -  0.         (h)  270°  +  0. 

4.  By  means  of  the  proper  formulas  and  the  table  on  page  161  find 

(a)  sin  (-  60°),  cos  300°,  cos  240°,  tan  315°. 

(b)  sin  330°,  cos  (-  120°),  cot  210°,  sec  150°. 

(c)  cos  (-  135°),  tan  120°,  tan  (-  150°),  sin  225°. 

5.  By  means  of  the  formulas  for  the  sine,  cosine,  and  tangent,  and  the 
reciprocal  relations,  derive  the  formulas  for  the  cotangent,  secant  and 
cosecant  of 

(a)  -  0;  (b)  90°  -  0;  (c)  180°  -  0;   (d)  180°  +  0;   (e)  360°  -  0. 

6.  Find  all  the  functions  of  142°.30;  of  118°.17. 

7.  Find  all  positive  values  of  0  less  than  360°  for  which 

(a)  cos  0  =  1;  tan  ^  =  1;  sin  ^  =  -  ^. 

(b)  sec  0 2;  cot  ^  =  -  1;  esc  0  =  \/2. 

(c)  sm  0  =  0.3486;  cos  0  =  -  0.8111;  tan  0  =  0.4770;  tan  ^  =  -  1.4770. 

8.  Find  all  the  values  of  0  for  which 

(a)  cos  ^  =  -  \/3/2;  cos  0  =  0.4761;  tan  0  =  2.     . 

(b)  sin  0  =  -  0.6460;  cos  0  =  0.5348;  tan  0  =  2.638. 

(c)  sec  0  =  1.4788;  esc  0  =  4.865;  cot  0  =  33.96. 

9.  Express  the  following  as  functions  of  0. 

(a)  sin  (0  -  90°). 

SoliUion.    Smce  ^  -  90°  -=  -  (90°  -  0),  we  have 

sin  {0  -  90°)  =  sm  [-  (90°  -  ^)]  =  -  sin  (90°  -  0)  =  -  cos  0. 

(b)  cos  {0  -  180°).      (c)  tan  {0  -  270°).         (d)  sin  (0  -  180°). 

10.  Construct  a  table  of  values  of  0  and  sin  0  for  values  of  0  taken  every 
10°  from  0°  to  360°,  expressing  0  in  radians  decimally  instead  of  in  terms 
of  TT  (see  Tables,  page  32),  and  giving  the  values  of  0  and  sin  0  to  two  deci- 
mal places.  Construct  the  graph  as  accurately  as  possible  from  this  table 
of  values. 

11.  As  in  the  preceding  exercise,  construct  a  table  of  values  and  draw 
the  graph  of 

(a)  cos  0f      (b)  tan  0,         (c)  cot  0,         (d)  sec.  0,  (e)  esc  0. 


TRIGONOMETRIC   FUNCTIONS 


199 


70.  Inclination  and  Slope  of  a  Straight  Line.  The  function 
tan  6  enables  us  to  express  precisely  the  relation  between  the 
slope  and  the  direction  of  a  line  (see 
page  52). 

The  upper,  right-hand  angle  (the 
northeasterly  angle)  which  a  line 
makes  with  the  a:-axis  is  called  the 
inclination  of  the  line. 

Parallel  lines  have  the  same  in- 
clination, and  conversely  (why?). 

Consider  any  line  through  the  origin.  Its  slope  m  is  the 
ratio  of  the  difference  of  the  ordinates  of  any  two  of  its  points 
to  the  difference  of  the  abscissas.  For  these  points  take  the 
origin  and  any  point  P{x,  y)  on  the  line,  above  the  x-axis.  We 
then  have,  in  either  figure. 


Fig.  116. 


t 


m  = 


y 


^  =  ^  =  tan  ^. 
0      X 


Hence  the  slope  of  the  line  through  the  origin  is  the  tangent 
of  the  inclination.  As  parallel  lines  have  the  same  slope  and 
the  same  inclination,  the  result  holds  for  any  hne.  We  thus 
have 


Fig.  116. 


Theorem  1.  The  slope  of  a  line  is  the  tangent  of  the  incUnor 
Hon,  i.e.,  m  =  tan  6. 

In  most  of  the  applications  of  the  linear  equation  y  =  mx  +  b, 
the  slope  is  of  greater  importance  than  the  incUnation,  be- 
cause the  slope  is  the  rate  of  change  of  y  with  respect  to  x. 


200 


ELEMENTARY  FUNCTIONS 


The  chief  importance  of  the  theorem  above  is  in  geometry. 
For  from  the  slope  the  direction  of  the  line  may  be  found, 
provided  the  same  unit  has  been  used  on  the  x  and  y-axes. 

Let  a  Une  start  in  a  horizontal  position  and  rotate  about 
one  of  its  points  through  half  a  revolution.  Its  inclination  d 
will  increase  from  0°  to  180°.  The  variation  of  tan  6  (see 
graph,  page  176)  shows  that  the  slope  of  the  Hne,  m  =  tan  dj 
will  increase  from  zero  through  all  positive  values,  and  become 
infinite  as  the  inclination  approaches  90°,  when  the  hne  be- 
comes vertical.  As  the  incHnation  increases  from  90°  to  180°, 
the  slope  is  negative,  and  its  numerical  value  decreases  to 
zero. 


Example  1.     Find  the  angle  formed  by  the  lines  I  and  V  whose  equa- 
tions are  3x  -  4y  +  12  =  0  and  2x  +  Z/  -  8  =  0. 

Solving  the  equations  of  the  lines  for  y 
we  get 


°i 

45 

1  ^      -b^ 

5^-^ 

Z  ^"^ 

i^^::^: 

>  » \ 

^ '.      5-« 

^S:         3::^- 

/"'  ^  ■  -  0   1  .  ^   ^    - 

y  =  to;  +  3,  and  y  =  -  2x  +  8 

and  hence  the  slopes  of  the  Hnes  are 

m  =  tan  ^  =  r  =  .75  and  m'  =  tan  6'  =  -  2. 
4 

From  the  tables  and  the  rule  in  Section 
69,  the  inclinations  of  the  lines  are 
e  =  36^87;   d'  =  180^  -  63^43  =  116°.57. 


Fig.  117. 

If  a  is  the  angle  between  the  lines,  then  a  =  6'  -  d  (why?).     Hence 

a  -  116**.57  -  36*'.87  =  79°.70. 

If  two  lines  are  perpendicular,  then,  using  the  notation  in 
Example  1,  a  =  90°,  so  that  d'  =  90°  +  d.     Hence 

m'  =  tan  6'  =  tan  (90°  +  0)  =  -  cot  ^  =  -  -^  =  - 1. 
^  tan  u         m 

That  is,  the  slope  of  one  is  the  negative  reciprocal  of  the 
slope  of  the  other.  The  converse  may  be  proved  by  retracing 
the  steps  in  the  reverse  order.     Hence  we  have 

Theorem  2.  Two  lines  are  perpendicular  if  and  only  if  the 
slope  of  one  is  the  negative  reciprocal  of  the  slope  of  the  other. 


ii 


TRIGONOMETRIC  FUNCTIONS  201 

EXERCISES 

1.  Find  the  equation  of  the  line  passing  through  the  point  (2,  3)  whose 
inclination  is  (a)  30°;   (b)  135*'. 

2.  Construct  each  of  the  lines  below,  and  find  its  inclination. 

(a)  2z-2y  +  7  =  0.       (b)  4a;  -  3^  -  12  =  0.       (c)  4x  +  3y  -12  =  0. 

3.  Find  the  angle  between  each  of  the  pairs  of  lines: 

(a)  VSx  -  32/  +  12  =  0  and  VSx  -  2/  -  3  =  0. 

(b)  5x -2y +  10  =  0  and  x  + Sy  -Q  =  0. 

4.  Find  the  angles  of  the  triangles  formed  by  the  lines  below.  How 
can  the  results  be  checked? 

(a)  X  +  y  -  4  =  0,  a;  -  Vdy  -3  =  0,  VSx  -y  -3  =  0. 
ih)2x-y  -Q  =  0,  x  +  2y  -3  =  0,  2x  +  3y  +  9  =  0. 

Definition.  The  angle  between  a  line  and  a  curve  at  a  point  of  inter- 
section is  the  angle  between  the  line  and  the  tangent  to  the  curve  at  that 
point.  The  angle  between  two  curves  at  a  point  of  intersection  is  the  angle 
between  the  tangents  to  the  curves  at  that  point. 

5.  At  what  point  on  the  graph  oi  y  =  x  -  x^  will  the  tangent  line  make 
an  angle  of  30**  with  the  x-axis?  Find  the  angles  at  which  the  curve  cuts 
the  X-axis. 

6.  Find  the  angle  made  with  the  x-axis  by  the  tangent  to  the  graph  of 
y  =  x^  -XB,t  the  point  for  which  x  =  1.  At  the  point  of  inflection  (Defini- 
tion, page  139). 

7.  Find  the  angle  between  the  graphs  of  x^  and  x^  at  the  point  (1,  1); 
between  the  graphs  of  a^  and  x^  at  the  same  point. 

8.  Find  the  points  on  the  graph  of  y  =  x^  -  x'^  at  which  the  inclination 
of  the  tangent  line  is  45**. 

9.  Find  the  angles  at  which  the  straight  line  y  =  4x  cuts  the  graph  of 
=  a:'  at  the  three  points  of  intersection. 

10.  Find  the  equation  of  the  line  through  the  origin  which  is  perpen- 
dicular to  the  tangent  to  the  parobola  y  =  x^  at  the  point  (1,  1). 


71.  Law  of  Sines.  A  theorem  of  geometry  states  that  if 
two  sides  of  a  triangle  are  unequal,  the  angles  opposite  them 
are  unequal  in  the  same  order,  and  conversely.  The  exact 
relation  between  the  sides  and  angles  is  given  by  the 

Law  of  sines.  The  sides  of  a  triangle  are  proportional  to  the 
sines  of  the  opposite  angles.    Symbolically ^ 

a b c 

sin  4  ~  sin  5  ~  sin  C 


202 


ELEIVLENTARY  FUNCTIONS 


Let  ABC  be  any  triangle,  and  draw  the  altitude  CD 
Then  in  the  right  triangles  ADC  and  BDC  we  have 

C 


h. 


Fig.  118. 

fe  =  6  sin  A  and  h  =  a  sin  B, 
whence  6  sin  A  =  a  sin  B, 

and  therefore 

a      _     b 
sin  A  ~  sin  5 
If  A  is  obtuse  (Fig.  1185),  then  ZCAD  =  180'*  -  A,  so  that 
/i  =.  6  sin  (180°  -  A)  =  6  sin  A     ((1),  page  192) 
as  before. 
In  like  maimer,  by  drawing  the  altitude  from  A  or  B,  we  get 
h  c  a  c  ,^. 

"= 5  =  - — n  01"   ~ 7  =  -= — 7y-  (1) 

Sin  B     sm  C       sin  A      sin  C 
72.  Law  of  Cosines.    This  law  replaces  the  two  theorems 
in  geometry  concerning  the  square  of  a  side  of  a  triangle  op- 
posite an  acute  angle  and  the  square  of  a  side  of  a  triangle 
opposite  an  obtuse  angle. 

Law  of  cosines.     The  sqitare  of  a  side  of  a  triangle  equals  the 
sum  of  the  squares  of  the  other  two  sides  less  twice  their  product 
times  the  cosine  of  the  included  angle.    Symbolically 
fl2  =  62  +  c2  -  26c  cos  A. 
62  =  c2  +  a2  _  2ca  cos  B. 
c2  =  a2  +  62  _  2a6  cos  C. 
If  A  is  acute  (Fig.  118A),  we  have 
a^^h^-\-  DB^ 

=  ¥  +  (c-  ADY        (since  DB  =  c  -  AD) 
^h^^c"-  2c'AD  +  AD^ 
s=  52  _|_  g2  _  2bc  cos  A 

(since  h^  +  AD^  =  ¥,  and  AD  =  6  cos  A). 


TRIGONOMETRIC  FUNCTIONS  203 

If  A  is  obtuse  (Fig.  USB),  we  have 

=^h^  +  (c  +  Any        (since  DB  =  c  +  AD) 
^h^  +  c'  +  2c'AD  +  AD^ 
=  62  +  c2  _  26c  cos  A 

(since  h^  +  AD^  -=  ¥   and  AD  ^  h  cos  (180°  -  A)  =  -  6  cos  A, 
(2),  page  192.) 

EXERCISES      • 

1.  Show  that  the  law  of  sines  reduces  to  the  first  two  of  formulas  (1), 
page  181,  if  C  =  90°.  What  does  the  third  form  of  the  law  of  cosines  be- 
come? 

2.  Prove  the  first  of  equations  (1),  Section  71,  if  B  and  C  are  both  acute. 

3.  Prove  the  second  of  equations  (1),  Section  71,  if  A  is  acute  and  C  is 
obtuse. 

4.  Prove  the  second  form  of  the  law  of  cosines  if  5  is  acute. 
6.   Prove  the  third  form  of  the  law  of  cosines  if  C  is  obtuse. 

6.  Find  the  ratio  of  two  sides  of  a  triangle,  a/6,  if 

(a)  A  =  40**  and  B  =  20°.     Is  one  side  double  the  other? 

(b)  A  =  60°  and  B  =  20°.  Is  sin  30  =  3  sin  67  Can  this  question  be 
readily  answered  from  the  graph  of  sin  6? 

7.  If  a  particle  is  in  equilibrium  imder  three  forces,  OA,  OB,  OC,  prove 

that  -: — 5777;  =  -; — TTTTf  =  -; — TrTfT     (The  Tcsultant,  OD,  of  OA  and  OB 
sm  BOC     sm  ADC     sm  AOB.  ' 

must  be  equal  and  opposite  to  OC,  and  hence  the  sides  of  the  triangle 

OAD  will  represent  the  forces  numerically.    Use  law  of  sines.) 

73.  Solution  of  Oblique  Triangles.  An  oblique  triangle  is 
one  none  of  whose  angles  is  a  right  angle.  The  constructions  of 
plane  geometry  show  that  an  oblique  triangle  may  be  con- 
structed if  three  of  its  six  parts  (sides  and  angles)  are  given, 
provided  that  at  least  one  of  them  is  a  side.  The  laws  derived 
in  the  last  two  sections  enable  us  to  solve  the  triangle,  that  is 
to  compute  the  three  unknown  parts  from  those  given.  It  is 
necessary  to  distinguish  four  cases,  according  as  there  are 
given: 

LA  side  and  two  angles. 

II.  Two  sides  and  the  angle  opposite  one  of  them. 
III.  Two  sides  and  the  included  angle. 
D  IV.   Three  sides. 


204  ELEMENTARY   FUNCTIONS 

The  first  two  cases  may  he  solved  by  the  law  of  sines,  and  the 
last  two  by  the  law  of  cosines.  The  last  two  may  also  be  solved 
by  using  first  the  law  of  cosines  and  then  the  law  of  sines. 

Case  I.  Given  a  side  and  two  angles,  the  third  angle  is  found 
from  A  +  B  +  C  =  1S0°,  and  the  other  two  sides  by  the  law  of 
sines. 

Example  1.  Two  forts  by  the  sea,  A  and  B,  are  12  miles  apart. 
At  A  the  angle  between  B  and  a  target  C  anchored  off  the  coast  is  37°.24, 
and  at  B  the  angle  between  A  and  C  is  42°.87.     Find  the  distance  from 

each  fort  to  the  target. 

We  are  given  a  triangle  determined  by 
A  =  37°.24,  B  =  42°.87,  c  =  12. 

We  have  C  =  180°  -  (A  +  B)  =  99°.69. 
From  the  law  of  sines, 

c  sin  A        J  -      c  sin  B 
a  =  — : — 77-j  and  0  =  — : — 7^> 
sm  C  sm  C 

,                  12  X  0.6052      „  „^-     .,  ,  ,      12  X  0.6803      ^  oot       1 

whence  a n  qq^j      "  ^-^^^  miles,  and  b  =  — ^       -      =  8.281  miles. 

Check.    Find  c  from  a,  6,  C.    By  the  law  of  cosines, 

c2  =  a2  +  62  -  2ab  cos  C 

-  7.372  ^  8.282  -  2  X  7.37  x  8.28  x  (-  0.168) 
=  54.4  +  68.6  +  20.5 
=  143.5, 
and  therefore  c  =  11.98. 

This  value  agrees  reasonably  well  with  the  given  value  of  c,  especially 
when  we  take  into  account  the  fact  that  the  table  of  squares  does  not  en- 
able us  to  use  all  four  figures  in  a  and  b. 

Case  II.  Given  two  sides  and  an  angle  opposite  one  of  them, 
the  angle  opposite  the  other  given  side  is  found  by  the  law  of  sines, 
then  the  third  angle  is  found  from  the  relation  A  -\-  B  -\-  C  =  180°, 
and  finally  the  third  side  is  found  by  the  law  of  sines. 

Example  2.  Two  straight  roads  diverge  at  an  angle  of  35°.  An 
automobile  starts  from  the  fork  in  the  road  and  runs  along  one  road  until 
a  cross  road  is  reached,  the  odometer  showing  the  distance  to  be  2.1  miles. 
It  turns  into  the  cross  road  and  after  running  1.4  miles  comes  to  the  other 
road.  How  far  is  the  automobile  from  the  starting  point,  and  at  what 
angles  does  the  cross  road  meet  the  other  two? 


TRIGONOMETRIC   FUNCTIONS 


205 


If  A  denotes  the  fork  in  the  road,  B  the  point  2.1  miles  from  A,  and  C 
the  third  point  reached,  a  triangle  is  determined  by  the  parts 

A  =  35^  c  =  2.1,  a  =  1.4. 

To  construct  the  triangle,  construct  angle  A  =  35°  with  one  side  hori- 
zontal, and  on  the  other  side  lay  off  AB  =  c  =  2.1.  Then  with  J5  as  a 
center,  describe  the  circle  with  radius 
o  =  1.4,  which,  in  this  problem,  cuts 
the  other  line  in  two  points  C  and  C. 
Either  BC  or  BC  may  represent  the 
cross  road,  so  that  the  problem  has 
iwo  solutions,  AC  and  AC.  These 
may  be  found  by  solving  the  triangles 
ABC  and  ABC 

To  find  C  we  have,  by  the  law  of 
sines, 

csinA      2.1x0.5736 


sin  C 


=  0.8610, 


a  1.4 

whence  C  =  59°.43  or  120°.57  (compare  Example  1,  page  197).    Smce  BCC 
is  isosceles,  C  =  ZAC'B'  =  180°  -  C, 

and  hence  C  =  59°.43  and  C  =  120°.57. 

Whence  B  =  180°  -  (A  +  C)  and  B'  =  180°  -  (A'  +  C) 

=  85°.57  =  24°.43 

To  find  b  =  AC  and  h'  =  AC,  we  have,  by  the  law  of  sines, 

in  AABC      6  =  ^^^      and      in  AAB'C     h' =  ^-^ 
sm  A  sin  A 

1.4x0.9969  1.4  X  0.4136 

~       0.5736  "       0.5736 

=  2.433  =  1.009 

Check.    By  the  law  of  cosines  we  have: 

In  triangle  ABC     c^  ^a^  +  b^  -  2ab  cos  C 

=  1.42  +  2.4332  -  2  X  1.4  X  2.433  X  0.5085 

=  1.960  +  5.91  -  3.464 

=  4.406,  and  hence  c  =  2.099. 

In  triangle  AB'C  c"^  =  a^  +  6^  -  2ab'  cos  C 

=  1.42  +  1.0092  _  1  X  1.4  X  1.009  X  (-  0.5085) 

=  1.960  +  1.018  +  1.437 

=  4.415,  and  hence  c  =  2.101. 

Both  of  these  values  agree  very  well  with  the  given  value  of  c.    In 
order  to  illustrate  the  method  of  solution,  the  computations  have  been 


206 


ELEMENTARY  FUNCTIONS 


carried  out  with  all  the  accuracy  permitted  by  the  tables.  Do  the  given 
data  warrant  us  in  saying  that  the  automobile  is  either  2.433  miles  or  1.009 
miles  from  the  fork  in  the  road? 

//  two  sides  of  a  triangle  and  the  angle  opposite  one  of  them  are 
given,  there  can  he  two  solutions  only  if  the  angle  is  acuie 
and  the  Me  opposite  it  is  less  than  the  other  given  side.  Under 
these  conditions,  there  will  be  two  solutions  unless  in  solving 
for  the  second  angle  (C,  in  Example  2),  it  is  found  that  its  sine 
is  equal  to  or  greater  than  unity  (sin  C^  1).  In  the  first  case 
there  is  but  one  solution,  a  right  triangle;  and  in  the  second 
case  there  is  no  solution,  since  the  sine  of  an  angle  cannot 
exceed  unity.  A  complete  statement  of  the  possible  solutions 
is  given  in  Exercise  3  below. 

Case  III.  Given  two  sides  and  the  included  angle,  the  third 
side  is  found  by  the  law  of  cosines,  and  then  the  remaining  angles 
hy  either  the  law  of  cosines  or  the  law  of  sines. 

Example  3.    To  find  the  distance  across  a  bay,  AB,  a  point  C  is  taken  j 
whose  distances  from  A  and  B  can  be  measured.     It  is  found  that  AC  is 
70  yards,  BC  is  120  yards,  and  Z  AC  Bis  50**.     Find  AB  and  also  the  angles 
at  A  and  B. 

A  triangle  is  determined  whose  given  parts  are  a  =  120,  6  =  70,  C  =  50**. 
By  the  law  of  cosines 

c2  =  a2  +  6^  -  2ab  cos  C 

=  120"*  +  702  -  2  X  120  X  70  X  0.6428 
=  14,400  +  4900  -  10,800 
=  8500,      whence      c  =  92.2. 
To  find  A  and  B,  we  use  the  law  of  cosines:  Substituting  the  values  of 

the  sides  in  a^  =  ¥  +  0^  -  26c  cos  A, 
we  get 

14,400  = 

4900  +  8500  -  2  X  70  X  92.2x  cos  A, 
whence        cos  A  =  -  0.0774 
and  hence 

A  -  180°  -  85^55  =  94^45. 

Substituting  the  values  of  the  sides  in 
fe2  =,  c2  ^  ci*  _  2ca  cos  B,  we  get 

8500  +  14,400  -  2  X  92.2  x  120  x  cos  B, 

0.8136 

35°.55. 


Fig.  121 

4900 

whence 

cos^ 

and  hence 

B 

TRIGONOMETRIC   FUNCTIONS  207 

Check.  The  angles  having  been  found  without  using  the  fact  that  the 
sum  of  the  three  angles  is  two  right  angles,  we  have 

A-{-B  +  C  =  94°.45  +  35^55  +  50**  =  180°. 

Having  found  A  as  above,  B  might  have  been  obtained  from 
A  +  B  +  C  =  180".  But  then  it  would  have  been  necessary  to  use  the 
law  of  sines  as  a  check,  and  little  would  have  been  gained. 

Case  IV.  Given  the  three  sides^  the  three  angles  are  found  by 
the  law  of  cosines,  or  one  may  he  found  by  the  law  of  cosines  and 
then  the  others  by  the  law  of  sines. 

Example  4.  The  lengths  of  a  triangular  lot  are  found  by  pacing  to 
be  80  feet,  50  feet,  and  100  feet.  Find  the  angles  at  the  corners  and  the 
area. 

A  triangle  is  determined  by  a  =  80,  6  =  50,  c  =  100.  Substituting  these 
values  in  the  three  forms  of  the  law  of  cosines  we  get 

802  =  502  +  1002  -  2  X  50  X  100  cos  A, 
502  =  1002  +  802  -  2  X  100  X  80  cos  B, 
100*  =  802  +  502  -  2  X  80  X  50  cos  C, 
and  hence         cos  A  =  0.6100,        whence        A  =  52*'.41 

(COS  B  =  0.8687,  B  =  29°.69 

cos  C  =  -  0.1375,  C  =  97°.90. 

Check :  A+B  +  C  =  180". 00. 

To  find  the  area,  draw  the  altitude  from  C  and  denote  it  by  h.    Then 
In  the  right  triangle  ACD,  h  =  b  ein  A  =  50  X  0.7924  =  39.62.    Then  the 
area   is   i  ch  =  50  x  39.62  =  1981    square 
feet. 

The  lengths  of  the  lines  in  the 
examples  preceding  and  in  the  fol- 
lowing exercises  have  been  chosen    ^  cioo 
with    but   one   or   two    significant                  p^^  122. 
figures,  in    order    to   simplify   the 

computations,  and  the  angles  to  four  significant  figures  for  the 
purpose  of  obtaining  drill  in  interpolation.  In  the  next 
chapter,  a  labor  saving  device  to  assist  in  the  computations 
will  be  considered,  and  further  exercises  given. 


208  ELEMENTARY   FUNCTIONS 


EXERCISES 

1.  In  Example  4,  find  B  and  C  by  the  law  of  sines.  What  point  in  the 
solution  might  be  overlooked,  which  one  is  more  likely  to  notice  in  using 
the  law  of  cosines? 

2.  Is  a  triangle  always  determined  if  values  are  given  for  a  side  and  two 
angles?    For  two  sides  and  the  included  angle?    For  the  three  sides? 

3.  If  two  sides,  say  a  and  6,  and  the  angle  opposite  one  of  them,  say  A, 
are  given,  show  that 

(a)  If  A  is  obtuse,  there  is  one  solution  if  a>5,  and  there  is  no  solution 
if  a^b. 

(b)  If  A  is  right,  there  is  one  solution  if  a  >  6,  and  there  is  no  solution 
if  a^6. 

(c)  If  A  is  acute,  there  is  one  solution  if  a  ^  6,  there  are  two  solutions  if 
6  sin  A  <a<b,  there  is  one  solution  if  a  =  b  em  A,  and  there  is  no  solution 
if  a<b  sin  A. 

4.  Determine  the  number  of  solutions,  by  Exercise  3,  if. 


(a)  a  =  5,           6  =  10, 

A  =  30°. 

(b)  a  =  5,           b  =  7, 

A  =  30°. 

(c)  6  =  10,          c  =  15, 

C  =  20°. 

(d)  a  =  7,           c  =  6, 

C  =  40°. 

Solve  the  following  triangles,  and  check  the  solution: 

(a)  B  =  62^74,  C  =  87^20 

,  o  =  10. 

(b)   6  =  7,           c  =  5. 

B  =  30°.17. 

(c)    6  =  7,           c  =  9, 

B  =  30M7. 

(d)  a  =  3,           6  =  7, 

C  =  120°. 

(e)   a  =  3,          6  =  7, 

c=5. 

(f)    o  =  10,        6  =  8, 

B  =  28°.16. 

6.  A  tunnel  is  to  be  built  through  a  hill  from  a  point  A  to  another  point 
B.  A  point  C,  at  the  same  level  as  A  and  B,  is  1000  feet  from  A  and  800 
feet  from  B  and  ZACB  =>  42°.     Find  the  length  of  the  tunnel. 

7.  A  man  starts  from  camp  and  walks  N.E.  for  5  miles,  and  then  22°.5 
east  of  south  until  he  reaches  a  point  from  which  the  camp  is  visible  in  a 
direction  due  west.     How  far  has  he  walked,  and  how  far  is  he  from  camp? 

8.  Two  batteries  of  artillery,  A  and  B,  are  four  miles  apart.  An 
enemies'  battery  is  located  at  a  point  C  such  that  ZBAC  =  64°.22  and 
ZABC  =  43°.17.     Find  the  range  for  each  battery. 

9.  A  field  is  bounded  by  two  roads  intersecting  at  right  angles,  and 
by  two  other  straight  lines.  Find  its  area  if  the  lengths  of  the  sides,  be- 
ginning at  the  corner  at  the  crossing  of  the  roads  and  measured  in  order 
around  the  field,  are  30,  60,  70  and  40  rods. 


TRIGONOMETRIC  FUNCTIONS  209 

10.  A  schooner  sails  10°  west  of  north  at  the  rate  of  7  knots  an  hour 
across  the  Gulf  Stream  at  a  place  where  it  flows  N.E.  with  a  velocity  of 
4  knots  an  hour.  Find  the  actual  velocity  of  the  schooner  in  direction 
and  magnitude. 

11.  The  current  in  a  river  flows  at  the  rate  of  2  miles  an  hour,  and  a 
man  rows  at  the  rate  of  4  miles  an  hour.  If  he  desires  to  cross  the  river 
at  an  angle  of  70°  with  the  bank,  in  the  direction  of  the  current,  in  what 
direction  should  he  row? 

12.  Solve  the  preceding  exercise  if  the  man  desires  to  cross  at  the  same 
angle  with  the  bank  but  in  the  upstream  direction. 

13.  Resolve  a  velocity  of  50  feet  per  second  into  two  components  in- 
clined at  10°  and  40°  respectively  to  the  direction  of  the  given  velocity. 

14.  A  boy  in  an  automobile  moving  40  feet  per  second  throws  a  ball  in 
a  horizontal  direction  inclined  at  50°  to  the  road  with  a  speed  of  30  feet 
per  second.    At  what  angle  to  the  road  will  the  ball  move? 

16.  A  road  runs  up  a  hill  at  an  angle  of  20°.  At  a  point  on  it,  500  feet 
from  the  foot,  the  angle  of  depression  of  a  horseman  on  the  road  leading 
to  the  hill  is  5°.     How  far  is  he  from  the  foot  of  the  hill? 

16.  To  find  the  width  of  a  river,  two  points  A  and  B  are  taken  on  one 
bank  100  feet  apart.  If  C  is  a  point  on  the  opposite  bank  such  that 
IBAC  =  62°.34  and  /.ABC  =  49°.82,  find  the  width  of  the  river. 

17.  At  a  certain  point  the  angle  of  elevation  of  the  top  of  a  mountain 
is  45°,  and  at  a  point  1000  feet  nearer  the  mountain  and  at  the  same  level 
as  the  first,  the  angle  of  elevation  is  54°.  13.     How  high  is  the  mountain? 

18.  A  ship  steams  due  east  at  the  rate  of  25  miles  an  hour,  and  the  smoke 
from  its  funnel  is  blown  in  a  direction  20°  south  of  west.  The  wind  gauge 
shows  an  apparent  velocity  of  35  miles  an  hour  for  the  wind.  Find  the 
actual  velocity  of  the  wind  in  direction  and  magnitude. 

74.  Inverse  Trigonometric  Functions.  To  find  the  inverse 
of  sin  X  (pages  40  and  114)  we  set  y  =  sin  x,  and  interchange 
X  and  y,  obtaining  x  =  sin  y.  The  solution  of  this  equation 
for  y  in  terms  of  x  requires  the  introduction  of  a  new  function 
which  is  called  the  angle  whose  sine  is  x,  and  which  is  denoted 
by  arc  sin  x.    Hence, 

if  jc  =  sin  y,  then  y  =  arc  sin  x. 

A  table  of  sines  may  be  regarded  as  a  table  of  angles  whose 
sines  are  given  (see  "  finding  6  if  sin  6  is  given  "  page  178). 
Thus  Example  2,  page  179,  might  have  been  stated:  find 
B  =  arc  sin  0.4332,  the  result  being 

arc  sin  0.4332  =  25°.76  +  n360°  or  154°.33  +  n360°. 


k 


210 


ELEMENTARY  FUNCTIONS 


arc  sin  x 


This  example  illustrates  the  fact  that  for  a  given  value  of  x 
arc  sin  x  has  not  only  one  but  a  boundless  number  of  values. 
This  is  apparent  from  the  graph  of  arc  sin  x,  which  is  sym- 
metrical to  that  of  sin  x 
with  respect  to  the  bi- 
sector of  the  first  and 
third  quadrants  (see  page 
114).  The  graph  also 
shows  that  the  function  is 
defined  only  for  values  of 
X  from  -  1  to  +  1  in- 
clusively. 

The  inverse  of  cos  x  is 
denoted  by  arc  cos  x  (read 
"the  angle  whose  cosine 
is  x");  of  tan  x,  by  arc 
tan  X,  etc. 
Definition.  The  principal  valve  of  any  ono  of  the  inverse 
trigonometric  functions  for  a  given  value  of  x  is  that  one  of  the 
boundless  number  of  values  of  the  function  which  is  smallest 
numerically.  If  two  values  of  the  function  are  equal  numerically, 
but  opposite  in  sign,  the  positive  value  is  the  principal  value. 

Unless  the  contrary  is  indicated, 
the  symbols  arasin  x,  arc  cos  x,  etc., 
will  he  used  in  this  work  to  denote  the 
principal  values  only. 

Thus        arc  sin  J  =  tt  /6, 


Fig.  123. 


and       arc  sin  (—  1)  =  —  7r/2. 

The  part  of  the  graph  which 
represents  the  principal  values  of 
arc  sin  x  is  given  in  the  figure. 

Inverse    trigonometric    functions 
are  of  much  importance,  although 
we  shall  use  them  but  little  in  this  course, 
venient  in  stating  a  general  result,  as  in  this 


FiQ.  124. 


They  are  con- 


TRIGONOMETRIC   FUNCTIONS 


211 


Example.    What  angle  is  subtended  at  the  center  of  a  circle  of  radius 
r  by  a  cliord  c  units  long? 

Choose  the  center  of  the  circle  as  the  origin  of  a  system  of  coordinates, 
and  let  the  ic-axis  be  perpendicular  to  the  chord.  Then  the  a>-axi8  bisects 
the  'chord  and  the  angle  formed  by  the 
radii  drawn  to  its  extremities.  From  the 
figure 

sin  d/2  =  (c/2)/r, 
whence         6/2  =  arc  sin  (c/2r) 
and  hence        6  =  2  arc  sin  (c/2r). 

If  c  =  3  and  r  =  5,  we  would  have  ^  =  2 
arc  sm  0.3  =  2  x  17*'.46  =  34°.92. 

The  notation  sin"^a;,  cos^^o;,  etc., 
is  used  sometimes  for  arc  sin  x,  arc 
cos  X.  etc. 


Fig.  125. 


EXERCISES 


1.  Construct  the  graph  of  arc  sin  x,  and  indicate  on  it  the  part  which 
represents  the  principal  values  of  the  function  for  all  possible  values  of  x. 
State  as  many  properties  of  the  function  as  can  be  readily  obtained  from 
the  graph. 

2.  Proceed  as  in  Exercise  1  for  the  function  (a)  arc  cos  x.     (b)  arc  tan  x, 

(c)  arc  cot  c.     (d)  arc  sec  x.     (e)  arc  esc  x. 

3.  Find  all  the  values  of  arc  sin  0;  arc  cos  ^;  arc  tan  (-1). 

4.  Find  the  value  (noting  the  convention  with  regard  to  principal 
values)  of 

(a)  arc  cos  (-  5);  arc  tan  2,  arc  sin  (-  0.3215). 
I'  (b)  arc  sin  f ;  arc  sec  2;  3  x  arc  sin  (-  VJ). 

6.  A  rope  I  feet  long  is  stretched  from  the  top  of  a  building  to  the 
ground,  the  lower  end  being  d  feet  from  the  building.  Find  a  general 
expression  for  the  angle  which  the  rope  makes  with  the  ground.  What 
is  the  angle  if  I  is  50  feet  and  c?  is  17  feet? 

6.  A  mountain  h  feet  high  is  viewed  from  a  point  d  miles  away  (hori- 
zontally). What  angle  does  the  line  from  the  point  of  observation  to 
the  peak  make  with  the  ground? 

7.  What  is  the  value  of  sin  (arc  sin  a)?    Of  arc  sin  (sin  a)? 

8.  Recall  the  method  of  solution  of  Exercises  9  and  10,  page  170.  Find 
the  value  of  (a)  sin  (arc  cos  ^).     (b)  tan  (arc  sin  ■^^).     (c)  cos  (arc  tan  -  2). 

(d)  tan  [arc  cos  (-  i)]. 

9.  If  the  maximum  distance  from  a  point  on  an  arc  of  a  circle  to  the 
chord  of  the  arc  is  d,  show  that  the  central  angle  subtended  by  the  arc  is 
2  X  arc  cos  (r  -  d)/r,  where  r  is  the  radius  of  the  circle. 


212  ELEMENTARY   FUNCTIONS 

MISCELLANEOUS  EXERCISES 

1.  If  a  creek  is  20  feet  wide,  and  if  from  a  point  4  feet  above  the  water's 
edge  on  one  side  the  angle  of  elevation  of  the  top  of  the  bank  on  the  other 
side,  directly  opposite,  is  7°.27,  how  high  is  the  bank? 

2.  An  object  weighing  60  pounds  is  supported  on  a  smooth  plane  whose 
inclination  is  60°  by  a  man  who  pushes  against  it  horizontally.  Find  the 
force  exerted  by  the  man  and  the  pressure  on  the  plane. 

3.  A  board  is  just  strong  enough  to  bear  an  object  weighing  100  pounds 
at  its  middle  point  when  the  board  is  supported  horizontally  at  its  ends. 
How  heavy  an  object  will  it  bear  at  its  middle  point  if  it  is  supported  at 
its  ends  in  a  position  inclined  at  30",  the  object  being  held  in  position  by 
a  rope  parallel  to  the  plane? 

4.  Solve  Exercise  3  if  the  object  is  held  in  position  by  a  man  pushing 
against  it  horizontally. 

5.  The  hatchway  into  the  hold  of  a  ship  is  16  feet  wide.  To  raise  an 
object  weighing  200  pounds  from  the  hold,  two  men  on  opposite  sides  of 
the  hatchway  pull  on  a  rope  which  passes  through  a  smooth  ring  fastened 
to  the  object.  Find  the  tension  of  the  rope  when  the  ring  is  6  feet  below 
the  deck.    What  force  does  each  man  exert?    Hint:  See  the  note  following. 

Note.  The  tension  of  a  cord  or  rope  which  passes  over  a  smooth  peg, 
or  over  a  pulley,  or  through  a  smooth  ring  is  assumed  to  be  the  same  on 
both  sides  of  the  peg,  pulley,  or  ring.  It  is  usually  not  the  same  if  the  rope 
is  tied  to  the  ring. 

6.  A  cord  is  tied  at  a  point  A,  passes  through  a  smooth  ring  B  weighing 
3  pounds,  over  a  pulley  C  at  the  same  height  as  A,  and  to  the  end  is  tied 
a  weight  of  2  pounds.  Find  ZBAC  when  the  ring  and  weight  are  in 
equiUbrium. 

7.  A  man  drags  a  trunk  across  a  room  by  pulling  on  the  handle  in  a 
direction  at  35**  to  the  horizontal.  If  the  trunk  weighs  150  pounds,  and 
if  the  friction  is  0.1  of  the  pressure  on  the  floor,  find  the  force  he  exerts 
at  the  instant  the  trunk  is  about  to  move.  Note  that  the  pressure  on  the 
floor  will  be  less  than  the  weight  of  the  trunk,  because  a  part  of  the  man's 
effort  tends  to  lift  the  trunk. 

8.  Find  the  force  exerted  in  pushing  the  trunk  in  Exercise  7,  when  the 
trunk  Is  just  on  the  point  of  moving,  If  the  man  pushes  down  on  the  trunk 
in  a  direction  inclined  at  25°  to  the  horizontal. 

9.  To  find  the  width  of  a  river,  a  point  A  is  taken  on  one  bank  directly 
opposite  a  tree  on  the  other  bank,  and  a  point  B  is  taken  100  feet  from  A 
In  the  line  of  the  tree  and  A.  At  B  the  angle  of  elevation  of  the  top  of 
the  tree  is  32°.  19,  and  at  A  it  is  4r.33.    Find  the  width  of  the  river. 

10.  If  a  body  on  a  rough  inclined  plane  is  just  on  the  point  of  moving 
down  the  plane,  show  that  the  coefficient  of  friction  is  equal  to  the  tangent 
of  the  angle  of  inclination  of  the  plane. 


TRIGONOMETRIC  FUNCTIONS  213 

11.  Rain  drops  are  falling  straight  down  with  a  velocity  of  20  feet  per 
second.  At  what  angle  would  they  appear  to  fall  to  a  man  walking  at 
the  rate  of  3  miles  per  horn*?  To  a  man  in  an  automobile  moving  at  the 
rate  of  20  miles  per  hour?  To  a  man  in  an  express  train  moving  at  the 
rate  of  60  miles  an  hour? 

12.  Prove  that  the  area  of  any  quadrilateral  is  equal  to  one-half  the 
product  of  the  diagonals  and  the  sine  of  the  angle  between  them. 

13.  The  wind  is  blowing  down  a  lake  5  miles  wide  at  a  rate  which  would 
blow  a  row  boat  a  mile  an  hour.  A  man  who  rows  at  the  rate  of  3  miles 
an  hour  desires  to  go  straight  across.  In  what  direction  should  he  row, 
and  how  long  will  it  take  him  to  cross? 

14.  Two  girls  hold  a  traveling  bag  weighing  40  pounds.  One  puUs 
on  the  handle  at  an  angle  of  10*  to  the  vertical,  the  other  at  15°.  What 
force  does  each  exert? 

15.  If  the  distance  an  ivory  ball  rolls  down  a  smooth  plane  inclined  at 
<  I  1,  2,  3,  4  5**  in  various  times  are  as  given  in  the  table, 
s  I  1.4,  5.5,  12.5,  22.2  the  units  being  feet  and  seconds,  find  the  ac- 
celeration due  to  gravity.  Analysis  of  the  problem:  Find  s  as  a  function 
of  t  from  the  table,  from  this  find  the  velocity  at  any  time,  then  the 
acceleration  with  which  the  ball  rolls  down  the  plane,  and  finally  the 
acceleration  due  to  gravity. 

16.  Charles'  law  states  that  the  rate  of  increase  of  the  volume  of  a  gas 
under  constant  pressure  per  degree  (Centigrade)  rise  of  temperature  is 
sJj  of  the  volume  at  0",  vq.    Hence  the  volume  v  at  any  temperature  d  is 

V  =  ^r;^  6  +  vo.    Boyle's  law  states  that  the  product  of  the  pressure  and 

volume  of  a  gas  at  constant  temperature  is  constant,  pv  =  k.  If  a  quantity 
of  oxygen  occupies  200  cubic  centimeters  at  a  temperature  of  17"  Centi- 
grade under  a  pressm-e  of  742  millimeters  (barometric  height),  find  its 
volume  at  0**  temperature  under  a  pressure  of  one  atmosphere  at  sea  level, 
or  760  millimeters.  Illustrate  graphically,  plotting  the  graphs  of  both 
laws  on  the  same  axes,  taking  v  on  the  vertical  axis  for  each  law.  First 
find  the  volume  Vo  under  a  pressure  of  742  millimeters;  the  graph  of 
Charles'  law  being  determined  by  the  given  data  and  the  fact  that  v  =  0 
when  ^  =  -  273**,  the  "  absolute  zero."  The  value  of  k  in  Boyle's  law  is 
determined  by  the  point  whose  coordinates  are  200  and  this  value  of  vo. 

17.  Construct  a  square  8  inches  on  a  side.  Its  area  is  64  square  inches. 
Cut  the  square  into  two  rectangles  of  widths  3  inches  and  5  inches.  Cut 
the  first  rectangle  along  a  diagonal  into  two  triangles.  Mark  points  on 
the  8  inch  sides  of  the  second  5  inches  from  opposite  vertices,  and  cut  along 
the  line  joining  them.  The  four  parts  of  the  original  square  may  be 
arranged  in  the  form  of  a  rectangle  whose  dimensions  are  5  inches  and  13 
inches,  and  whose  area  Is  65  square  inches.    Explain  the  fallacy. 


CHAPTER  V 

EXPONENTUL  AND   LOGARITHMIC   FUNCTIONS 

75.  Introduction.    The  formula  for  the  nth  term,  I,  of  a 

geometrical  progression  whose  first  term  is  a,  and  whose  ratio 

is  r,  is 

I  =  ar^-\  (1) 

If  we  are  given  a  =  3,  r  =  2,  Z  =  96,  then  n  may  be  found. 
Substituting  the  given  values,  we  have 

96  =  3  X  2«-S 
whence  2"-^  =  32  =  2^,  (2) 

and  hence  n  -  1  =  5, 

from  which  n  =  6. 

Equation  (2)  differs  from  most  of  the  equations  arising  in 
algebra,  in  that  the  unknown  n  appears  in  an  exponent.  The 
possibiHty  of  finding  n  by  elementary  methods  is  due  to  the 
fact  that  32  is  recognized  as  a  power  of  2. 

If  we  set  a  =  3  and  r  =  2  in  (1),  we  get 

l  =  3x  2--K  (3) 

Regarding  n  as  variable,  this  equation  defines  I  as  a  func- 
tion of  n.  For  integral  values  of  n,  I  is  the  nth  term  of  a 
geometrical  progression,  but  in  studying  I  as  a  function  of  n 
we  need  not  restrict  n  to  integral  values.    Thus  if  n  =  |, 

Z  =  3  X  23-1  =  3  X  2*  =  3  X  1.414  =  4.242. 

Thus  the  value  of  I  may  be  found  readily  by  algebraic  methods 
for  many  fractional  values  of  n.    But  if  n  is  irrational,  for  ex-  | 
ample,  if  n  =  \/2  or  w  =  tt,  we  cannot  compute  I  by  algebraic  j 
methods.     Hence  Z  is  a  transcendental  function  of  n  (definition,  ! 
page  39).  i 

214  i 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS   215 

Definition.  An  exponential  function  is  one  in  which  the 
variable  occurs  in  an  exponent.  For  example,  equation  (3) 
defines  I  as  an  exponential  function  of  n. 

The  simplest  form  of  an  exponential  function  is  6*,  where  6 
is  a  constant,  other  than  unity,  called  the  base.  It  has  the 
property,  in  common  with  more  general  forms  of  exponential 
functions,  that  if  x  increases  in  arithmetical  progression,  the 
function  increases  in  geometrical  progression.  For  example, 
values  of  a;  in  the  table  are  in  arithmetical  ^progression,  since 

successive  values  of  Ax  are  equal,  while 
a;  I  1      2      3      4  ^       » 

•^-\    '      '      ' — j^     the  values  of  2^  are  in  geometrical  pro- 
gression, since  the  ratio  of  any  value  to 
the  preceding  is  2. 

Exponential  functions  are  important  because  in  many  fields, 
especially  in  physics,  one  variable  changes  (increases  or  de- 
creases) in  geometrical  progression  as  another,  frequently 
time,  changes  in  arithmetical  progression.  The  former  is 
always  an  exponential  function  of  the  latter.  Examples  of 
such  changes  are  Newton's  law  of  cooling  of  a  heated  body; 
the  variation  of  atmospheric  pressure  with  the  altitude;  the 
law  of  chemical  reaction;  and  the  law  of  organic  growth.  In 
these  examples,  and  generally  in  nature,  the  change  proceeds 
continuously.  The  amount  of  a  sum  of  money  at  compound 
interest  is  an  exponential  function  of  the  time,  but  the  changes 
come  at  stated  intervals  instead  of  continuously.  However, 
the  b\!lsiness  of  some  very  large  firms,  which  make  many  loans 
a  day,  approximate  a  condition  of  compounding  interest  every 
instant.  And  this  property  of  an  exponential  function  is 
often  called,  following  Lord  Kelvin,  the  compound  interest  law. 
Another  name  which  has  been  suggested  is  the  snow-hall  law. 

The  inverse  of  the  function  6*  is  called  the  logarithmic  func- 
tion, and  a  table  of  values  of  this  function  is  spoken  of  as  a 
table  of  logarithms.  Such  a  table  is  an  invaluable  labor-saving 
device.  It  enables  us  to  replace  the  laborious  processes  of 
multipHcation  and  division  by  the  simpler  operations  of  addic- 
tion and  subtraction.  It  also  makes  it  possible  to  reduce 
the  very  tedious  operations  of  computing  powers  and  roots  to 


216 


ELEMENTARY  FUNCTIONS 


multiplication  and  division,  and  through  these  to  addition  and 
subtraction. 

For  this  labor-saving  device  we  are  indebted  to  John  Napier 
(1550-1617),  who  arrived  at  it  by  considering  a  function  which 
increased  in  arithmetical  progression  as  the  variable  decreased 
in  geometrical  progression.  Following  Napier,  this  tool  was 
put  in  a  more  serviceable  form  by  Henry  Briggs  (1561-1631). 

We  shall  use  a  table  of  logarithms  for  simplifying  many 
computations,  especially  in  connection  with  the  solution  of 
triangles. 

The  introduction  of  the  exponential  and  logarithmic  functions 
completes  the  list  of  functions  to  be  studied  in  this  course 
(see  Classification,  page  38). 

76.  Graph  of  the  Exponential  Function  b'^j  b>l.  In  the 
following  table  of  values  of  the  exponential  function  2",  the 
value  of  the  function  for  a;  =  0  is  obtained  by  means  of  the 
definition  6^  =  1,  and  the  value  for  any  negative  value  of  x  is 


obtained  by  means  of  the  definition  ft"**  =  j--    By  the  first  defi- 
nition, we  have  2P 


1,  and  by  the  second  2'^  ^  02  '^  4* 


2x 


2,  ~  1,  0,  1,  2,  3 
h       h,  1,  2,  4,  8 


The  table  of  values  is  readily  com- 
puted and  the  graph  plotted  as  usual. 
The   process  is  so  simple   that  it  is 
hardly  worth  while  to  discuss  the  graph  in 
advance.     The   following   properties  of   the 
graph  are  apparent. 

1.  The  intercept  on  the  y-axis  is  1. 

2.  The  graph  Hes  entirely  above  the  x-axis. 

3.  The  X-axis  is  an  asymptote. 

4.  Any  fine  parallel  to  the  y-axis  cuts  the 
graph  once  and  once  only. 

5.  The  graph  rises  to  the  right  more  and 
more  rapidly. 

The  form  of  the  graph  of  the  exponential  function  6^,  6>1, 
is  very  much  like  that  of  2'.  As  indicated  in  Fig.  127,  the 
graph  always  cuts  the  y-axis  at  2/  =  1,  Hes  entirely  above  the 


m 

g  -y- 


Fig.  126. 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    217 


• 

y 

T 

— 

g 

t&    3' 

i' 

1 

n_j 



1  / 

JT/ 

it/" 

fjl 

0    i 

uy 

J 

9 

1 

A 

..  .,  0| 

1      2 

1      J,. 

X 

FiQ.  127. 


X-axis,  which  is  an  asymptote,  and  runs  up  to  the  right.  If 
b>2,  the  curve  rises  more  rapidly  to  the  right  of  the  y-axis 
than  in  the  case  of  2*,  and  the  larger  the 
value  of  b  the  more  rapid  is  the  rise. 

The  values  of  b  of  most  importance  in 
mathematics  are  10  and  a  number  denoted 
by  "  e"  whose  value  to  four  figures  is 
6  =  2.718.  It  was  proved  in  1844  that  there 
are  numbers  which  are  not  the  roots  of  any 
algebraic  equation  no  matter  how  high  the 
degree,  and  the  first  number  definitely 
proved  to  be  of  this  sort  was  e,  in  1873. 
In  1882  this  same  fact  was  proved  about 
TT,  and  by  means  of  this  it  was  proved  that 
it  is  impossible  to  "  square  the  circle  '*  with  ruler  and  compass. 

A  table  of  values  of  e^  is  given  in  Huntington's  Tables, 
page  30. 

77.  Properties  of  the  Exponential  Function  6',  6>1.  The 
properties  of  the  function  which  correspond  to  the  above 
properties  of  the  graph  are  as  follows: 

1.  For  all  exponential  functions  6^  =  1. 

2.  The  exponential  function  6*  is  positive  for  all  real  values 
of  X. 

3.  As  X  decreases  indefinitely  through  negative  values  6* 
approaches  0  as  a  limit. 

4.  For  each  value  of  x  there  is  one  and  only  one  value  of  6=^. 

5.  The  function  b^  increases  as  x  increases  and  the  rate  of 
change  of  b'  also  increases.  Of  two  exponential  functions  the 
one  with  the  larger  base  has  the  greater  rate  of  change  for 
x>0. 

Other  important  properties  of  the  function  which  are  not  so 
apparent  from  the  graph  are  the  following: 

6.  If  the  values  of  x  be  chosen  in  arithmetical  progression, 
the  corresponding  values  of  the  function  are  in  geometrical 
rrogression. 

Let  the  values  of  x  be 

a,  a  +  d,  a  i-  2d,  a  -\-  Sd  .  .  , 


218  ELEMENTARY  FUNCTIONS 

Then  the  values  of  the  function  are 

6«,  6°+^,  ¥+^,  ¥+^,  .  .  . 

which  form  a  geometrical  progression,  for  any  one  of  these 
values  may  be  obtained  from  the  preceeding  by  multiplying 
byt*^. 


7.  6*"  •  fe'^  =  6"*+". 

8. 

9.    (b*")"  =  fe*"". 

10. 

\/b^  =   h^/n 

Properties  7,  8,  9,  proved  in  elementary  algebra  for  integral 
values  m  and  n  of  x,  are  true  for  all  values  of  x.  The  defini- 
tion 10  holds  also  for  all  values  of  x. 

If  fix)  =  6*,  the  last  four  properties  may  be  written,  in  the 
reverse  order: 

11.  f(m  +  n)  =  6»«+»  =  6"»6«. 

12.  f(m  -n)  =  &♦«-"  =  6«/6«. 

13.  f{mn)  =  6"*"  =  (fc"*)". 


"•  <t)  - 


In  these  forms,  the  analogy  of  these  relations  respectively 
to  (1),  (2),  (3),  and  (4)  on  page  153  is  apparent.  Thus  seem- 
ingly unrelated  rules  of  elementary  algebra  are  in  reahty  anal- 
ogous properties  of  the  functions  6*  and  x"". 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    219 


78.  Computation  by  means  of  an   Exponential  Fmiction. 

The  computations  in  the  following  examples  are  considerably 
simplified  by  means  of  the  adjoining  table 
and  properties  7,  8,  9,  10  of  the  exponential 
function. 

Example  1.   Find  the  product  (0.0078125)  (1048576). 

From  the  table  we  have  0.0078125  =  2"^. 
1048576  =  2^0 
Whence,  by  property  7,  the  product   =  2-^+20  =  2^'. 
Therefore  (0.0078125)  (1048576)  =  8192. 

Example  2.    Find  the  quotient  32768/524288. 
From  the  table  we  have   32768  =  2^. 
524288  =  2^9. 
By  property  8,        the  quotient  =  2^-^^  =  2^. 
Therefore,  32768/524288  -  0 .  0625. 

Example  3.    Find  ^. 
Since  8  =  2',  we  have  8^  =  (2')*. 
By  property  9,  =2^. 

Therefore  8«  =  262144. 


X 

2* 

-7 

0.0078125 

-6 

0.015625 

-5 

0.03125 

-4 

0.0625 

-3 

0.125 

-2 

0.25 

-1 

0.5 

0 

1 

1 

2 

2 

4 

3 

8 

4 

16 

5 

32 

6 

64 

7 

128 

8 

256 

9 

512 

10 

1024 

11 

2048 

12 

4096 

13 

8192 

14 

16384  . 

15 

32768 

16 

65536 

17 

131072 

18 

262144 

19 

524288 

20 

1048576 

Example  4.    Find  v/1048576. 


Since  1048576=  22°,  </1048576 

By  property  10 

Therefore  -M048576 


220/6  =  2*. 
16. 


inverse  function. 


These  examples  are  sufficient  to  show  that 
an  extensive  table  would  shorten  very  appre- 
ciably the  labor  of  finding  a  product,  quotient, 
power,  or  root.  In  practice  it  is  found  more 
convenient  to  use  a  table  of  values  of  the 

EXERCISES 


1.  Plot  the  graphs  of  2*,  c*,  and  3*  on  the  same  set  of  axes.  (See  page  30 
of  the  Tables  for  values  of  e*.)  Calculate  for  each  function  the  value  of 
Ay /Ax  for  the  intervals  -  2  to  -  1,  -  1  to  0,  0  to  +  1,  +  1  to  +  2,  and  de- 
termine which  has  the  greatest  value  of  Ay /Ax  and  which  the  least  for 
each  interval. 

2.  Plot  the  graph  of  2"*  from  a  table  of  values.  In  what  two  ways  is 
the  graph  related  to  that  of  2*? 


220  ELEMENTARY  FUNCTIONS 

3.  How  does  3~*  change  as  x  increases  in  arithmetic  progression? 

4.  Plot  the  graphs  of  the  following  fimctions  on  the  same  axes 
2*,  (I)*,  (1)*,  (f)*)  (§)*•  What  two  relations  exist  between  the  graphs  of 
the  second  and  the  fourth  function?  How  does  the  function  6*  vary  if 
6  >  1?  if  6  =  1?  if  6  <  1? 

6.  Plot  the  graphs  of  e*,  e~*,  i(e*  +  e~*)  and  i(e*  -  e"*)  on  the  same  axes 
for  the  range  x  =  -2toa;  =  +2at  intervals  of  0.5.     (See  Tables,  p.  30.) 

The  functions  i  (e*  +  e~*)  and  \  (e*-  e~*  ),  have  somewhat  the  same  rela- 
tion to  the  equilateral  hyperbola  that  the  trigonometric  functions  have  to 
the  circle.  They  are  called  respectively  the  hyperbolic  cosine  and  hyper- 
bolic sine  and  are  symbolized  by  cosh  x  and  sinh  x, 

6.  If  x',  y'  and  x",  y"  are  two  sets  of  values  satisfying  the  equation 
y  =  lf,  show  that  the  value  of  the  function  corresponding  to  the  arithmetic 
mean  of  x'  and  x"  is  the  geometric  mean  of  y'  and  y". 

7.  Find  the  value  of  2*  for  x  =  1.50,  using  Exercise  6  and  the  values  for 
a;  =  1  and  x^2.  Then  find  the  value  of  2'  for  x  =  1.25;  for  x  =  1.75. 
Arrange  the  results  in  tabular  form. 

8.  Referring  to  the  table  in  Exercise  7,  between  what  two  values  of  x 
does  X  =  \/2  lie?  The  value  of  2^-*^  is  an  approximate  value  of  2^.  Find 
it  by  applying  Exercise  6  several  times,  using  the  table  in  Exercise  7, 
and  choosing  the  successive  values  of  x'  and  x"  so  that  their  arithmetic 
mean  approaches  1.41. 

9.  By  means  of  the  table  in  Section  78  find  the  value  of  each  of 
the  following: 

(a)  (524288)  (0.015125).         (b)  4096/0.0078125.  (c)  16*. 

(d)  V16384.  (e)   (V'(32768)  (0.0625) /5 12)'.  (f)  VW^- 

10.  A  glass  marble  falls  from  a  height  of  4  feet  and  rebounds  one-half  the 
distance  fallen,  falls  again  and  rebounds  one-half  the  preceding  distance 
fallen,  and  so  on.  Express  the  distance  fallen  each  time  as  a  function 
of  the  number  of  times  it  has  fallen  and  draw  the  graph.  How  far  does 
it  fall  the  eighth  time? 

11.  If  the  planets  are  numbered  in  the  order  of  their  distance  from  the 
sun  the  distance  from  the  sun  to  the  nth  planet  is  approximately 
4  +  3(2)'*~2^  the  distance  of  the  earth  being  represented  by  10.  Compare 
the  results  of  substituting  in  this  formula  with  the  following  table. 

Planet  Mercury      Venus      Earth      Mars      Asteroids      Jupiter 

True  distance         3.9  7.3  10         15.2  27.4  52 

Saturn  Uranus        Neptune 

95.4  192  300 

If  there  is  a  planet  external  to  Neptune,  at  what  approximate  distance 
may  we  expect  it  to  be  found? 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    221 

12.  Show  for  the  exponential  function  e*  that  the  average  rate  of  change 

^w  /gAx—  \\ 

is  -^  =  e*  I  —7 1°    Find  the  value  of  the  second  factor  if  Ax  =  0.1,  0.01. 

Ax         \    Ax     I 

Assuming  that  the  limit  of  the  second  part  is  1  as  Ax  approaches  0, 
find  m  for  the  exponential  e*  and  determine  the  angle  that  the  graph  makes 
with  the  2/-axis. 

13.  Plot  the  graph  oi  y  =  ke"*^,  first  letting  k  =  1  and  choosing  m  suc- 
cessively equal  to  -  2,  -  1,  0,  +  |,  +  1,  +  2,  and  then  letting  m  =  I  and 
choosing  k  successively  equal  to  -  1,  -h  5,  +  2.  How  do  changes  in  m 
and  k  affect  the  graph? 

14.  The  pull,  P,  needed  to  check  a  weight,  W,  being  lowered  by  means  of 
a  rope  wrapped  around  an  iron  drum,  is  given  by  the  equation  P  =  We""^, 
where  the  coefficient  of  friction  is  m  =  0.3,  and  d,  the  angle  of  contact  of 
the  rope  with  the  drum,  is  measured  in  radians.  If  W  =  500  pounds  plot 
the  graph  of  P  as  a  function  of  d.  What  is  the  value  of  P  if  the  rope  is 
wrapped  4  times  around  the  drum? 

79.  The  Logarithmic  Function,  the  Inverse  of  the  Exponential 
Function.  To  find  the  inverse  of  2^,  let  2/  =  2^  and  interchange 
X  and  y,  obtaining  x  =  2^.  The  solution  of  this  equation  for 
y  in  terms  of  a;  is  a  new  function  called  the  logarithm  of  x  to  the 
base  2,  and  it  is  denoted  by  log2  x. 

Thus  X  =  2y       and        y  =  log2  x 

are  forms  of  the  same  equation,  in  the  one  case  solved  for  x 
and  in  the  other  for  y.    In  general,  we  have  the 

Definition.  If  6"*  =  n,  then  m  is  said  to  be  the  logarithm 
of  n  to  the  base  b. 

Hence  the  logarithm  of  a  number  to  the  base  b  is  the  exponent 
of  the  power  to  which  b  must  be  raised  to  equal  the  given  number. 

It  is  frequently  convenient  to  change  from  the  exponential 
to  the  logarithmic  form  of  this  relation,  and  vice-versa. 

For  example,  from  3^  =  243,  we  have  logs  243  =  5. 

Again,  to  find  logs  625,  we  let  logs  625  =  m,  whence  5"*  =  625. 

Since  5^  =  625,  we  have  m  =  4  and  hence  logs  625  =  4. 

80.  Graph  of  the  Logarithmic  Function.  The  graph  of 
y  =  log2  X  may  be  readily  obtained  from  that  of  2*  by  the  rule 
in  Section  39,  page  113,  and  from  it  certain  properties  of  the 
graph  are  obvious. 


222 


ELEMENTARY  FUNCTIONS 


"'    -/- 

.-l.l\l 

-l^  ^  ^ 

-   /  / 

:    //    ^-T;r, 

^            «, 

^<^    '■'  ^ 

.|  ,^    -4    .   J   ^    W. 

±  / 

1.  The  intercept  on  the  a;-axis  is  a:  =  1. 

2.  The  graph  Ues  entirely  to  the  right  of  the  y-axis. 

3.  The  2/-axis  is  an  asymptote. 

4.  The  graph  is  above  the  x-axis 
to  the  right  of  a:  =  1  and  below  the 
ic-axis  to  the  left  of  a;  =  1. 

5.  The  graph  rises  to  the  right  but 
at  a  decreasing  rate. 

6.  A  Une  parallel  to  the  a;-axis  with 
an  intercept  on  the  y-axis  at  y  =  1 

Fia.  128.  cuts  the   graph   at   a   point   whose 

abscissa  is  x  =  2. 
As  indicated  in  Fig.  129,  which  shows  the  graph  of  log5  x 
for  6  =  2,  3,  10;  the  graph  of  any  logarithmic  function, 

y  =  logftX,  6>1, 

cuts  the  a;-axis  at  x  =  1 ,  lies  entirely  to  the  right  of  the  y-axis, 

which  is  an  asymptote,  lies  above 

the  a;-axis  to  the  right  of  the  line 

a;  =  1,  and  below  to  the  left  of 

a:  =  1,  rises  at  a  decreasing  rate 

as  it  moves  to  the  right,  and  is 

cut  by  the  line  ?/  =  1  at  a  point 

P  whose  abscissa  is  b. 

81.  Properties    of    the    Log-  -piQ.  129. 

arithmic  Function,  log6  x,  b>l. 

Corresponding  to  the  properties  of  the  graph  are  the  following 
properties  of  the  function: 

1.  For  any  base,  logb  1=0.    For  6^  =  1. 

2.  Logb  a;  is  a  real  number  for  positive  values  of  x  only. 

3.  As  x  approaches  0,  logb  x  approaches  -  00,  that  is, 
log6  0  =  -  00,  6>1. 

4.  The  logarithm  of  a  positive  number  greater  than  unity 
is  positive.  The  logarithm  of  a  positive  number  less  than 
unity  is  negative. 

5.  The  function  log6  x  increases  and  the  rate  of  change  of 
log6  X  decreases  as  x  increases,  and  of  two  logarithmic  func- 


y  ■ 

1 

^-^ 

"^ 

lolfsX         I 

^=d 

■ 

1 

'> 

~~ 

""■ 

'4.-  1 

"/ 

•    4    i 

1 

' 

f  1  " 

7 

f 

r 

r 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    223 


tions,  the  one  with  the  larger  base  has  a  smaller  rate  of  change 
fora;>l. 

6.  The  logarithm  of  the  base  itself  is  unity.    For  since 
h^  =  h,  we  have  logb  6  =  1. 

Other  properties  of  the  function  are  easily  deduced  from  the 
corresponding  properties  of  the  exponential  function  as  follows: 

7.  Theorem.     The  logarithm  of  the  product  of  two  numbers  is 
equal  to  the  sum  of  the  logarithms  of  the  numbers. 

Let  p  =  h"",        whence        log6  p  =  my 

and  q  =  6",         whence        log6  q  =  n. 

Then  pq  =  h"'b''  =  6'"+". 

Therefore  logb  pq  =  m  +n  why? 

=  log6  p  +  logb  q. 

"    8.   Theorem.     The  logarithm  of  the  quotient  of  two  numbers  is 
equal  to  the  logarithm  of  the  dividend  minus  the  logarithm  of  the 
divisor. 
Let  p  =  b'^  and  q  =  6",  whence  Iog6  p  =  m  and  log6  q  =  n. 
Then  p/q  =  &""/&"  =  &"*""• 

Therefore  logb  pjq  =  m  -  n  why? 

»  =  logft  p  -  log6  q. 

^    9.   Theorem.     The  logarithm  of  the  nth  power  of  a  number 
equals  n  times  the  logarithm  of  the  number. 

Let  p  =  6"*,        whence        log^  p  =  m. 

Then  p"  =  (6"*)'*  =  6"*". 

Hence  logb  p"  =  mn  why? 

=  n  logb  p. 

10.   Theorem.     The  logarithm  of  the  nth  root  of  a  number 
equals  the  logarithm  of  the  number  divided  by  n. 

Let  p  =  6"*,    whence    logb  p  =  m. 

Then  v^p  =  v^6^  =  fc"'/". 

Hence      logb\^p  =  m/n         why? 

=  -  logb  p. 


224  ELEMENTARY  FUNCTIONS 

If  }{x)  =  logftX,  properties  7,  8,  9,  10,  may  be  written 

Jijpq)  =  logft  vq  =  logb  J)  +  log6  q,  (11) 

Jijplq)  =  logft  p/g  =  logo  p  -  logb  g.  (12) 

}{jp^)  =  logb  ^"=71  logb  p.  (13) 

/(\^p)  =  log5  <^V=];  log6  p.  (14) 

These  properties  are  respectively  analogous  to  those  of  x" 
given  by  (3),  (4),  (5),  (6),  page  153. 

If  the  first  two  be  written  in  the  reverse  order,  we  have 

f{v)  +fiQ)  =  logfe  p  +  logb  q  =  logb  pq,  (15) 

and  /(p)  -fiq)  =  log?,  p  -  logb  q  =  log^  p/g,  (16) 

which  are  analogous  to  (7)  and  (8),  page  153. 

By  means  of  the  theorems  in  7  and  9  the  solutions  of  Ex- 
amples 1  and  3,  page  219,  may  be  written  as  follows: 

Example  1.  log2  0.0078125  =  -  7  Example  3.  log2  8=3 

log2     1048576  =    20  loga  8«  =  6  logoS 

loga  (product)    =13  =18 

Therefore                    product    =    8192  Therefore                   8«  =  262144 

EXERCISES 

1.  Express  the  following  exponential  equations  in  logarithmic  form. 

23  =  8,  3^  =  81,  4-2  =  3^6,  10-3  =  0.001,  16i  =  \/2. 

2.  What  are  the  logarithms  of  1,  2,  16,  1024,  0.125  with  respect  to  the 
base  2?    Express  the  answers  in  exponential  and  logarithmic  form. 

3.  Find  logz  2048,  logs  81,  loge  2.718,  log^  9,  log^  16.    Express  the 
answers  in  exponential  form. 

4.  Find  lo&,  6^,  b^^^',  log2  (log4  16),  loge  (1/e),  b^^'f^''^^^''. 

5.  (a)  Change  the  equation  log«  6=  -kt  +  log«  ^o  to  the  form  ^  =  ^o  e~^, 

(b)  Show  that      -  ^lo©,  (l/x^)  =  log,  x. 

(c)  Find/(a:)  if  \ogbf{x)  =  log^  (l  -  x)  -  21og6  x  -  ^logj.  (1  +  x). 

6.  Plot  the  graphs  of  log2  x,  loge  x,  logs  x,  on  the  same  set  of  axes.  (See 
page  31  of  the  Tables  for  values  of  log*  x.)  Calculate  the  value  of  Ay /Ax 
for  each  function  for  the  interval  1  to  2. 

7.  Plot  the  graph  of  logi/aX.  In  what  two  ways  can  this  graph  be 
obtained  from  that  of  log«  x? 

8.  Illustrate  each  of  the  theorems  in  7,  8,  9,  10,  Section  81,  on  the 
graph  of  log2  z. 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    225 

9.  Write  the  solutions  of  Examples  2  and  4,  Section  78,  in  logarithmic 
forms. 

10.  Show  by  Theorems  7  and  8  that  logt  (  — )  =  loe>  V  +  log&  3  -  logt  **• 

11.  Using  the  table  m  Section  78  find  (0.03125)  (262144)/ (32768). 

12.  Prove  the  theorem,  if  the  numbers  x,  x',  x",  etc.,  are  in  geometric 
progression,  their  logarithms  are  in  arithmetic  progression. 

13.  Plot  the  graphs  of  logs  x,  logs  x^,  logaVx  on  the  same  axes.  How 
can  the  graphs  of  the  last  two  be  obtained  from  that  of  the  first  function? 

14.  Plot  the  graph  oi  y  =  log2  {x  +  k)  for  the  values  fc  =  0.  1,  -  1  on  the 
same  axes,  and  discuss  the  change  effected  in  the  graph  by  a  change  in  A;. 

16.  The  equation  for  the  economic  law  of  diminishing  utiUty  is 
y  =  k  logb  (x/c),  where  x  denotes  income;  y  happiness;  c  the  income  suf- 
ficient for  necessities;  and  A;  is  a  constant  depending  on  individuality. 
Plot  a  graph  of  the  law,  determine  the  graphical  significance  of  c  and  A;, 
and  discuss  the  law  for  x^c. 

82.  Common  Logarithms.  Logarithms  to  the  base  10  are 
called  common  logarithms.  The  first  table  of  logarithms  to 
the  base  10  was  published  in  1617  by  Henry  Briggs.  It  con- 
tained logarithms  of  nmnbers  from  1  to  1000. 

In  writing  logarithms  to  the  base  10,  the  base  is  usually 
omitted.  Thus  y  =  logio  x  is  written  y  =  log  Xy  which  means 
that  X  =  10^.    That  is 

The  common  logarithm  of  a  number  is  the  power  to  which  10 
mitst  he  raised  to  obtain  the  number. 

A  four-place  table  of  common  logarithms  of  numbers  from  1 
to  10  is  given  on  pages  16  and  17  in  Huntington's  TableSc 
The  graph  of  this  table  of  logarithms  is  shown  in  the  figure. 

The  method  of  inter- 
polation is  the  same  as       t/ 
in    the    tables    already 
used. 

For  example,  from  the 
tables  we  find  log  2.342 

=  0.3696    which    means  '  Fjq^  13q^ 

that  100-3698  =  2.342. 

The  number  2.342,  and  its  logarithm,  the  number  0.3696, 
are  the  coordinates  of  a  point  P  on  the  graph. 


226  ELEMENTARY  FUNCTIONS 

Either  from  the  graph  or  from  the  table  we  see  that 

The  logarithm  of  any  nmnber  between  1  and  10  is  a  positive 
decimal  fraction  less  than  1. 

We  proceed  to  develop,  by  illustration,  the  rule  for  finding 
from  the  table  the  logarithm  of  any  positive  number  <1  or 
>  10. 

If  we  translate  the  exponential  equations  10-^  =  o.Ol, 
10-1  =  0.1,  W  =  1,  W  =  10,  102  =  100,  W  =  1000,  into  the 
equivalent  logarithmic  equations  log  0.01  =  -  2,  log  0.1  =  -  1, 
log  1  =  0,  log  10  =  1,  log  100  =  2,  log  1000  =  3,  we  see  that 

The  Ipgarithm  of  an  integral  power  of  10  is  an  integer. 

Since  23.42  =  (2.342)  (100,  we  have,  using  the  theorem  in 
7,  Section  81,  log  23.42  =  log  (2.342)  (10^  =  log  2.342  + 
log  101  =  0.3696  +  1  =  1.3696. 

Similarly 

234.2  =  2.342-102,  whence  log  234.2  =  log  2.342  flog  W^  2.3696, 
2342.  =  2.342.  lO^,  whence  log  2342  =  log  2.342  +  log  10^  =  3.3696, 
0.2342  =  2.342-  lO'S  whence  log  0.2342  =  log  2.342  +  log  10-^  = 

0.3696  -  1, 
0.02342  =  2.342-10-2,  whence  log  0.02342  =  log  2.342  +  log  lO-^ 

=  0.3696-2. 


The  values  of  x  considered  above  have  the  same  sequence  of, 
digits  2,  3,  4,  5  and  the  corresponding  values  of  log  x  have  th 
same  decimal  part  0.3696. 

Multiplication  of  2.342  by  an  integral  power  of  10  mere! 
moved  the  decimal  point  without  altering  the  sequence  of 
digits,  and  corresponding  to  this  operation,  the  logarithm 
0.3696  was  increased  or  diminished  by  an  integer  without 
altering  the  decimal  part. 

Any  positive  number  x<l  or  >10  can  be  expressed  in  the 
form  X  =  x'-lO**  where  re'  is  a  number  in  the  range  l<x<10, 
and  w  is  an  integer,  positive  or  negative. 

Hence  log  x  =  log  x'  +  log  10"  =  log  x'  -f  n,  where  log  x\ 
is  a  positive  decimal  fraction  <1  which  can  be  obtained  fro 
the  table  of  logarithms  and  n  is  an  integer. 

Hence,  we  see  that 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    227 

1.  The  common  logarithm  of  a  positive  number  consists 
of  two  parts,  a  decimal  fractional  part  and  an  integral  part. 

2.  The  decimal  part,  called  the  mantissa,  depends  only  on 
the  sequence  of  digits  of  the  number  and  is  obtained  from  the 
table  of  logarithms  of  numbers  from  1  to  10. 

3.  The  integral  part,  called  the  characteristic,  is  determined 
by  the  position  of  the  decimal  point  in  the  number  with  refer- 
ence to  the  position  immediately  following  the  first  significant 
digit. 

4.  Moving  the  decimal  point  n  places  to  the  right  (or  left) 
in  the  number  is  equivalent  to  increasing  (or  diminishing)  the 
logarithm  by  n. 

To  find  the  logarithm  of  a  number  outside  the  range  of  the 
table  we  proceed  as  in  Examples  1  and  2. 

i Example  1.    Find  log  678.4 
I  log  678.4  =  log  (6.784)  (lO^)  =  log  6.784  +  2. 

rrom  the  table,  log  6.784  =  0.8315. 

Therefore  log  678.4  =  2.8315. 

Example  2.     Find  log  0.0004867. 

The  number  4.867,  lying  between  1  and  10,  has  the  same  sequence  of 
digits  as  the  given  number. 

From  the  table  log  4.867  =  0.6872. 

The  given  number,  0.0004867,  is  obtained  from  4.867  by  moving  the 
decimal  point  4  places  to  the  left.    Hence  by  property  4  above, 

log  0.0004867  =  0.6872  -  4. 

The  solution  of  Example  1  emphasized  the  reasoning  in- 
volved in  finding  the  characteristic,  while  in  Example  2  we 
have  apphed  the  rule  in  4.  Except  for  purposes  of  explanation 
there  is  no  need  of  writing  anjrthing  but  the  result. 

To  find  a  number,  given  the  logarithm  of  the  number,  proceed 
as  in  Examples  3  and  4. 

Example  3.    Find  x  given  log  x  =  2.5137. 

Disregarding  the  characteristic  2  for  the  moment,  we  find  from  the  table 
that  the  number  whose  logarithm  is  0.5137  is  3.264. 
Therefore  x  =  (3.264)  (lO^)  =  326.4. 
Example  4.    Find  x  if  log  x  =  0.8174  -  3. 


228 


ELEMENTARY  FUNCTIONS 


We  find  from  the  table  that  the  number  whose  logarithm  is  0.8174 
6.567. 

Since  the  characteristic  of  the  given  logarithm  is  3  less  than  that  of 
0.8174,  the  required  number  x  is  obtained  from  6.567  by  moving  the  deci- 
mal point  3  places  to  the  left  (see  rule  4  above).     Hence 

X  =  0.006567. 


The  graphical  significance  of  the  characteristic  and  mantissa 
is  shown  in  the  following  figures,  in  which  the  scales  on  the 
axes  are  chosen  differently  for  the  sake  of  clearness. 

The  first  figure  represents  the  graph  of  log  x  for  the  range 
10^37^100,  with  the  scale  on  the  a;-axis  much  reduced. 

AP  is  the 


';i 


2/A 


M 


W 

p  ^ - — 

— 

u 

1 " 

[0^96 

v 

3 
A 

+1 

1 

[)      23 

.42 

100       X 

A 


y 

A 

.1      .2342                                                    ] 

L 

1j 
-1- 

^ 

o.Gm^.^^ — 

0.3696 

W"x 

U"  I 

i                                                    I 

ni 

B 
Fig.  131. 


IS  tne  log- 
arithm of  OA,  AB 
is  the  characteris- 
tic, and  BP  the 
mantissa.  The 
characteristic  AB 
of  the  logarithm 
of  any  nimiber  OA 
in  this  range  is 
+  1,  while  the 
mantissa  BP  is  the 
same  as  that  of 
the  number  in  the 
range  1  ^  a;  ^  10  with  the  same  sequence  of  digits.  The  por- 
tion JJ'V'W  corresponds  to  the  portion  UVW  of  the  figure  in 
the  first  part  of  the  section. 

The  second  figure  represents  the  graph  of  log  x  for  the  range 
0.1  ^  a;  ^1,  with  the  scale  on  the  x-axis  much  enlarged.    The 
portion    U"V''W"  corresponds  to  the  portions    U'VW  and 
UVW  of  the  other  two  figures. 

We  have  from  the  table  log  0.2342  =  -  1  +  0.3696,  which  is 
represented  in  the  figure  by  AP  =  AB  +  BP. 

If  the  subtraction   indicated   is   performed   the   logarithm 
becomes  AP  =  -  0.6304. 

But  in  this  form  of  the  logarithm  the  rule  connecting  the 
mantissa  and  sequence  of  digits  in  the  number  is  not  the  same 


i 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    229 

as  that  for  numbers  greater  than  unity.  In  order  to  retain  this 
relation,  the  subtraction  indicated  is  not  performed,  and  the 
logarithm  is  written  in  the  form  0.3696  -  1,  or  9.3696  -  10, 
or  with  the  new  symbol  1.3696,  thus  retaining  the  positive 
mantissa  0.3696  and  its  connection  with  the  sequence  of  digits 
2,  3,  4,  2. 

It  is  to  the  simpHcity  of  the  relations  between  the  mantissa 
of  the  logarithm  and  the  sequence  of  digits  in  the  number, 
the  characteristic  of  the  logarithm  and  the  position  of  the 
decimal  point  in  the  number,  that  the  system  of  common 
logarithms  owes  its  superiority  to  all  other  systems  for  purposes 
of  computations.  (See,  for  example,  the  rule  for  shifting  the 
decimal  point  in  the  system  of  logarithms  to  the  base  e,  given 
on  page  31  of  the  Tables.) 

83.  Computation  by  means  of  Common  Logarithms.  The 
value  of  a  product,  quotient,  power  or  root  may  be  found  by 
logarithms  by  means  of  the  Theorems  7,  8,  9,  10  in  Section  81. 
The  following  examples  will  illustrate  the  methods. 

To  find  a  product  by  means  of  logarithms. 

Example  1.  Find  the  distance  around  the  earth  at  the  equator,  if 
the  equator  is  regarded  as  a  circle  of  radius  3963. 

We  have  C  =  2r  3963,  whence  by  Theorem  7,  log  C  =  log  2  +  log  ir 
+  log  3963.  Before  turning  to  the  table  of  logarithms,  write  out  a  blank 
form  as  indicated  on  the  left.  When  this  is  filled  in  it  gives  the  computa- 
tion on  the  right. 

log  2=  log  2  =  0.3010 

log  TT  =  log  7r  =  0.4971 

log  3963  =  log  3963  =  3.5980  We  have  from  the  table, 

log  C  =  log  C  =  4. 3961  if  log  x  =  0. 3961 

C=  0  =  24890  a;  =  2.489 

To  find  a  quotient  by  means  of  logarithms. 

Example  2.  Find  x  =    (0.003468)7(0.4783). 

iWe  have,  by  means  of  Theorem  8,  Section  81,  log  x  -  log  0.003468 
log  0.4783. 

k  log  0. 003468  =  0. 5400  -3  =  1. 5400  -  4 

f  log  0. 4783      =  0. 6797  -  1  =  0.6797  -1 
I  log  x  =  0.8603 -3 

^  a; -0.007250 


230  ELEMENTARY  FUNCTIONS 

The  mantissa  of  the  numerator  being  less  than  the  mantissa  of  the 
denominator,  we  add  1  to  and  subtract  1  from  the  characteristic  of  the 
numerator  in  order  to  avoid  a  negative  mantissa  in  the  difiference. 

If  two  measurements  are  made  to  four  figures  their  product 
or  quotient  should  not  contain  more  than  four  significant 
figures  (Section  26).  The  product  or  quotient  may  be  obtained 
by  a  four-place  table  of  logarithms  to  the  correct  number  of 
significant  figures.  In  general,  the  use  of  a  table  of  loga- 
rithms in  computing  products  and  quotients  conforms  to  the 
rule  for  the  nimiber  of  significant  figures,  if  the  number  of  sig- 
nificant figures  in  the  measurements  agrees  with  the  number 
of  decimal  places  in  the  logarithms. 

To  find  a  power  by  means  of  logarithms. 

Example  3.    Find  x  =  (0.008964)<. 
By  Theorem  9,  Section  81,  log  x  =  4  log  (0.008964) 
log  0.008964  =  0.9525-    3 

4 

log  X  =  3. 8100  -  12  =  0. 8100  -  9 
re  =  0.000000006457 

To  find  a  root  by  means  of  logarithms. 

Example  4.    Find  x  =  (0.6785)^.    By  Theorem  10,  log  x  =  f  log  0.6785. 
log      0.6785  =  0.8315-1 
4 
\na  (0  6785")*  =  o  qoBO  -  4        '^^^  characteristic  is  altered  in  order  to 
3  12  3260  -  3     ^htain  an  integral  characteristic  upon  divid- 

log  X  -  0. 7753  -  1     ^°^  ^^  ^'  *^^  ^°^®^  ^^  *^®  ^^^^' 
X- 0.5962 

EXERCISES 

1.  Compute  the  values  of  the  following  expressions  by  means  of  loga- 
rithms. In  each  case,  write  a  blank  form  for  the  computation  before 
turning  to  the  tables. 

(a)  (3.462)  (23.14).        (b)  4795/2439.        (c)  32.86".       (d)  VsiO 

(e)   ^Q-'^'^^^  [f^-^^^  (f)    ^32.96/12.78  (g)  V41.26V324.6 

^.oo9 

(h)  56.34/973.4.  Suggestion:  Write  the  characteristic  of  the  numerator 
in  the  form  (3  -  2). 

(i)  -^.03764.  Suggestion:  The  characteristic  of  the  given  number  may 
be  written  in  the  form  (1  -  3). 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    231 

2.  Find  the  distance  to  a  tower  59.47  feet  high  from  a  point  from  which 
the  angle  of  elevation  of  the  top  of  the  tower  is  21'*.87. 

3.  Find  the  area  of  an  isosceles  triangle  whose  base  is  2L5  inches  and 
whose  vertex  angle  is  27°.  16.  Note  that  it  is  sufficient  to  find  the  logarithm 
of  the  altitude  and  not  the  altitude  itself. 

4.  Apportion  a  claim  for  $35.55  damages  on  freight  which  traveled  on 
the  M.C.  428  miles,  N.Y.C.  175  miles,  I.C.  235  miles,  if  each  road  pays 
in  proportion  to  the  number  of  miles  traveled  on  it. 

5.  Find  the  cost  of  1000  rivets  weighing  4  ounces  each,  the  output  be« 
ing  at  the  rate  of  78  per  hour,  if  the  cost  of  raw  material  is  $4.50  a  ton,  if 
labor  is  $0,375  an  hour,  and  the  overhead  expense  is  $0,123  an  hour. 

6.  What  is  the  cost  of  674j  pounds  of  chromic  acid  95  %  pure,  if  acid 
90  %  pure  costs  7  cents  a  lb.? 

7.  A  cellar  230  feet  by  330  feet  by  15.5  feet  is  excavated  by  means  of  a 
scoop  with  a  capacity  of  |  cubic  yards  which  makes  15  trips  an  hour. 
Estimate  the  cost  of  drawing  away  the  dirt  if  a  driver  and  team  cost  $3.75 
a  day  of  10  hours  and  draw  two  loads  of  33^  cubic  feet  each  time. 

8.  The  average  daily  circulation  of  a  newspaper  for  six  semi-annual 
periods  from  1912  to  1915,  expressed  in  hundred  thousands,  was  210, 
229,  230,  246, 260  and  298.  Find  the  percentage  of  increase  in  circulation 
for  each  period  over  the  preceding. 

9.  Solve  the  following  quadratic  equations  using  logarithms  wherever 
allowable. 

(a)  2.13x2  +  4.76a;  _  3.32  =  Q.  (b)  32.6x2  -  87.5a;  +  43.7  =  0. 

10.  Find  the  radius  of  a  parallel  of  latitude  through  a  point  whose 
latitude  is  45°  N.,  assuming  the  radius  of  the  earth  to  be  3963. 

11.  The  area  of  a  sphere  in  terms  of  the  radius  r  is  given  by  the  equa- 
tion S  =  47rr2.  Find  the  area  of  the  surface  of  the  earth  considered  as  a 
sphere  of  radius  3963:miles. 

12.  The  volume  of  a  sphere  in  terms  of  the  radius  is  F  =  f  Trr'.  Find 
the  volume  of  the  earth  (see  11).  If  the  average  density  of  the  earth  is 
5.2,  and  the  weight  of  a  cubic  foot  of  water  is  62.4,  find  the  weight  of  the 
earth  in  tons. 

13.  Find  the  area  of  the  orbit  of  the  earth  assuming  it  to  be  a  circle  of 
radius  92,800,000  miles.  How  many  miles  does  the  earth  travel  in  a  day 
If  we  assume  a  year  to  be  equal  to  365  days? 

14.  Kepler's  third  law  of  planetary  motion  states  that  for  different 
planets  the  squares  of  the  times  of  describing  their  orbits  are  proportional 

to  the  cubes  of  the  mean  distances  from  the  sun.    That  is    m^  =  T^3* 

Given  the  earth's  mean  distance,  D  =  92,800,000,  its  periodic  time  T  =»  365, 
and  the  periodic  time  of  Mars  t  =  686,  find  the  mean  distance  of  Mars 
from  the  sun. 


232 


ELEMENTARY  FUNCTIONS 


15.  If  the  distance  of  the  earth  from  the  sun  changed  from  93,000,000 
to  90,000,000,  assuming  Kepler's  third  law,  show  that  the  year  would  be 
shortened  by  about  17.6  days.  (Let  T  =  365.3.)  Since  no  appreciable 
diminution  in  the  year  has  been  noted  from  ancient  times  to  the  present, 
what  inference  can  be  drawn? 

16.  In  railroad  surveying  a  simple  curve  is  defined  as  a  circular  arc 
joining  two  tangents. 

The  degree  of  curve,  D,  is  the  angle  at  the  center  of  the  circle  subtended 
by  a  chord  of  100  feet. 

(a)  Find  the  degree  of  curve  if  the 
radius  is  800  feet. 

(b)  Find  the  radius  if  the  degree  of 
curve  is  5°. 

(c)  Find  the  length  of  the  arc  AB  oi  a. 
4°  curve  if  the  central  angle  B  =  28*^. 

(d)  The  exterior  angle  at  V  is  52**,  and 
the  tangent  distance,  T,  is  to  be  800  feet. 
Find  the  radius  and  the  degree  of  curve. 
How  far  will  the  curve  pass  from  the 
vertex,  F? 

(e)  Find  T  if  it  is  required  that  the 
degree  of  curve  shall  not  exceed  4"  for 

Fig.  132.  two  tangents  with  an  external  angle  of 

40°. 

17.  If  a  body  is  constrained  to  move  in  a  curved  path,  a  force  directed 
outward  arises  which  is  called  the  centrifugal  force.  For  example,  as  a 
train  rounds  a  curve  the  pressure  of  the  flanges  of  the  wheels  against  the 
outer  rail  of  the  curve  is  a  centrifugal  force.    The  magnitude  of  this  force 

in  pounds  is  C  = ,  where  W  =  weight  in  pounds,  v  =  velocity  in  feet 

gr 

per  second,  r  =  radius  of  the  circle  in  feet  and  g  is  the  acceleration  due  to 

gravity  =  32.2. 

(a)  An  automobile  weighing  2  tons  rounds  a  curve  whose  radius  is  600  J 
feet  at  a  velocity  of  25  miles  an  hour.  What  is  the  magnitude  of  the  force] 
tending  to  make  it  skid? 

(b)  The  weight  of  a  mass  situated  at  the  equator  of  the  earth  is  de- 
creased by  the  centrifugal  force  due  to  the  rotation  of  the  earth  on  its  axis. 
Find  this  decrease  for  a  weight  of  500  pounds.  What  would  be  the  decrease 
in  latitude  60°  N.? 

(c)  At  what  angle  should  a  circular  automobile  speedway  one  mile  in 
circumference  be  banked  for  a  speed  of  100  miles  an  hour  in  order  that 
there  shall  be  no  tendency  to  skid? 

18.  During  the  war  with  Spain  in  1898,  a  chain  letter  was  started  for 
the  benefit  of  the  Red  Cross.    The  person  starting  it  wrote  to  10  friends 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    233 

numbering  each  letter  1 ;  each  of  these  was  to  write  to  10  friends  numbering 
their  letters  2,  etc.  The  chain  was  to  be  completed  by  letters  numbered 
100.  If  the  chain  had  not  been  broken  how  many  letters  would  have 
been  numbered  100?  Approximately  how  many  letters  in  all  would  there 
have  been?    How  does  this  compare  with  the  population  of  the  world? 

84.  Solution  of  Triangles.  The  formulas  (1),  page  181,  and 
the  law  of  sines  involve  multipHcation  and  division  only,  and 
hence  the  computations  in  the  solution  of  right  triangles  and  in 
Cases  I  and  II  of  oblique  triangles  may  be  effected  expeditiously 
by  means  of  logarithms. 

The  law  of  cosines  involves  addition  and  subtraction  and 
hence  is  not  well  adapted  to  logarithms.  In  Chapter  VIII 
formulas  will  be  derived  by  means  of  which  the  computations 
in  the  solution  of  Cases  III  and  IV  of  obUque  triangles  can  be 
effected  by  means  of  logarithms. 

Example.  To  find  the  distance  from  a  point  A  on  the  shore  of  a  bay 
to  a  point  B  off  shore,  a  point  C  is  taken  on  the  shore  350  feet  from  A. 
The  angles  BAC  and  BCA  are  found  to  be  84**.  13  and 
72°.76.    Find  AS.  }P 

We  have  6  =  350.  Log  6  =  2.5441 

A  =  84M3.  log  sin  C  =  0.9800-1. 

C  =  72°.76.  3.5241-1. 

Then        B  =  180^  -  (A  +  C)  log  sin  B  =  0.5939-1. 

=  23°.ll  Log  c  =  2.9302 

&sinC  c  =  851.7. 


And 


sin  B 


The  logarithms  of  sin  B  and  sin  C  may  be  obtained     A      b^SSO 
1  directly  from  page  20  of  the  Tables,  Fig.  133. 

Checking  the  solution  of  a  triangle.  In  any  numerical  compu- 
tation it  is  of  the  utmost  importance  that  the  accuracy  of  the 
results  be  checked  in  some  way. 

If  ABC  is  a  right  triangle,  an  excellent  check  is  given  by  the 
^  Pythagorean  theorem,  c^  =  a^  -\-  h^,  which  can  be  adapted  to 
logarithmic  computation  by  writing  it  in  the  form 


a  =  Vc2  -  62  =  V(c  -  6)  (c  -}-  6).  (1) 

Notice  that  the  possibility  of  expressing  c^  -  6^  as  a  product 
is  the  basis  of  the  adaptation  to  the  use  of  logarithms,  and  that 


234  ELEMENTARY  FUNCTIONS 

the  property  of  x^  used  here  is  a  special  case  of  (8),  page  153. 
In  general,  if  the  difference  of  two  values  of  a  function  /(x), 
namely,  /(a)  -  /(6),  can  be  expressed  as  a  product,  this  property 
of  the  function  enables  us  to  compute  the  difference  by  means 
of  logarithms. 

An  excellent  check,  which  is  simpler  than  the  law  of  cosines, 
for  the  solution  of  an  oblique  triangle  by  the  law  of  sines  is 
given  by  the  relation 

c  =  a  cos  B  +  6  cos  A.  (2) 

This  may  be  established  from  the  figures  in  Section  71, 
page  202,  by  the  use  of  right  triangles.  Similar  expressions  may 
be  written  for  a  and  h. 

EXERCISES 

1.  By  means  of  the  relations  sec  A  =  1/cos  A  and  esc  A  =  1/sin  A, 
show  how  log  sec  A  and  log  esc  A  can  be  found. 

2.  Solve  the  following  triangles  and  find  their  areas. 

(a)  a=    13.75,    5=    76^23,  0  =  90**. 

(b)  a  =  243.7,       c  =  431.2,      C  =  90^ 

(c)  A  =    34M6,  5=    92°.  86,    6  =  32.68. 

(d)  a=    14.96,     6=    12.32,    5  =  49°.  17. 

3.  A  level  road  runs  directly  toward  a  hill.  From  the  top  of  the  hill 
the  angles  of  depression  of  two  milestones  on  the  road  are  12°.17  and 
9**.48.     Find  the  height  of  the  hill  in  feet. 

4.  Two  points  A  and  B,  325  feet  apart,  are  situated  on  the  edge  of  a 
canyon.  At  A  the  horizontal  angle  between  B  and  a  point  C  across  the 
canyon  and  at  the  bottom  is  76*'.23.  At  B,  the  horizontal  angle  between 
A  and  C  is  64°.  18  and  the  angle  of  depression  of  C  is  69°.32.  Find  the 
depth  of  the  canyon. 

5.  Express  the  area  of  a  regular  polygon  of  n  sides  as  a  function  of  n 
and  of  the  side  a.  Use  this  formula  to  determine  the  area  of  a  regular 
pentagon  whose  side  is  34.8  inches. 

6.  Show  that  the  area  of  a  quadrilateral  va  A  =  \  dd'  sin  B,  where 
d  and  d'  are  the  diagonals  and  B  is  the  angle  between  them. 

7.  A  farm  has  the  shape  of  a  quadrilateral  ABCD,  BC  =  256  yards, 
ZABC  =  112°,  ZDBC  =  38°.5,  ZACB  =  42°,  ZDCB  =  124°.  Find  the 
selling  price  at  $85  an  acre.     (See  Exercise  6.) 

8.  Two  points  A  and  B  are  on  opposite  sides  of  a  river  1250  yards 
wide,  A  being  450  yards  farther  upstream  than  B.  In  what  direction 
relative  to  the  bank  should  an  auto  boat  be  pointed  in  order  to  proceed  in 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    235 

a  straight  line  from  B  to  A,  if  the  velocity  of  the  stream  is  4.6  miles  an  hour 
and  that  of  the  boat  is  16.3  miles  an  hour?  How  long  will  it  take  the  boat 
to  make  the  journey? 

9.  The  angle  of  elevation  of  a  cliflF  750  feet  high  viewed  from  a  ship 
due  east  was  18''.26.  The  ship  sailed  southwest  and  from  a  second  point 
of  its  course  the  angle  of  elevation  of  the  cliff  was  22°.06.  Find  the  dis- 
tance between  the  two  points  of  observation.  What  point  of  the  course 
of  the  ship  was  nearest  to  the  cliff? 

10.  In  order  to  extend  a  base  line  beyond  an  obstacle  which  could  be 
sighted  across,  a  surveyor  chose  four  stations,  as  indicated  in  the  figure. 
The  following  measurements 

were  made:  ^  j         ^    ^  \^      (. 

AB  =  U5.6  feet,  ZBAD 
=  32°.63,  ZCBD  =  43^88, 
ZBZ)C  =  114^76.    Find  BC. 

11.  A  monument  stands 
on  the  top  of  a  hill.  From  a 
certain  point  on  the  level 
ground  below  the  hill,  the  angles  of  elevation  of  the  top  and  the  base  of 
the  monument  are  28''.37  and  25°.63,  while  at  a  point  325.7  feet  nearer 
the  hill  the  angle  of  elevation  of  the  top  of  the  monument  is  30**.32.  Find 
the  height  of  the  monument. 

12.  To  find  the  horizontal  distance  to  and  the  height  of  an  observation 
balloon  A  anchored  behind  the  enemies  lines,  a  base  line,  J5C,  478  feet 
long  is  laid  off  and  the  horizontal  angles  between  A  and  C  at  B,  and  be- 
tween B  and  A  at  C  are  found  to  be  74''.38  and  83''.27.  At  B  the  angle 
of  elevation  of  A  is  17°.85.    Find  the  distance  and  the  height  desired. 


ASTRONOMICAL  EXERCISES 

The  zenith  of  a  point  A  on  the  surface  of  the  earth  is  the  point  in  the 
heavens  in  line  with  A  and  the  center  of  the  earth. 

The  zenith  distance  of  a  heavenly  body  with  respect  to  a  point  A  is  the 
angle  which  a  line  to  the  body  makes  with  the  line  to  the  zenith  at  A. 

1.  At  a  point  A,  a  star  in  the  zenith  is  chosen,  and  a  distance  AB  is 
measured  along  the  arc  of  a  meridian  to  a  point  B  where  the  zenith  dis- 
tance of  the  star  is  1**.  If  the  length  of  the  arc  AB  is  69.4  miles  find  the 
radius  of  the  earth.     (See  Fig.  135.) 

(In  view  of  the  great  distance  of  the  star  from  the 'earth,  what  assump- 
tion can  be  made  with  respect  to  the  lines  AS  and  BS,  and  hence  what 
magnitude  may  be  assigned  to  the  angle  A0B1) 

2.  The  north  latitude  of  Berlin  is  52**  23',  and  the  south  latitude  of 
Cape  of  Good  Hope  is  33**  5'.  Find  the  chord  distance  between  the  two 
places.    Take  r  =  3963.     (See  Fig.  136.) 


236 


ELEMENTARY  FUNCTIONS 


3.  At  Berlin  and  Cape  of  Good  Hope  the  zenith  distance  zx  and  zi  of 

the  moon  were  measured  simultaneously  and  found  to  be  21  =  53°  10'  1", 

zi  =  33°  37'  32".     Find  the 
distances  BM  and  OM. 

4.  When  the  moon  is  in 
the  position  where  we  see 
haK  the  illuminated  portion 
(i.e.  half-moon)  the  angle  at 
the  moon  is  90°.  At  such  an 
instant  the  angle  at  E  was 
measured  and  found  to  be 
89.86°.  Calculate  the  dis- 
tance to  the  sun  using  the 
distance  to  the  moon  as  a 
base  hne. 

6.  The  annual  parallax  of 
a  star  is  the  angle  at  the  star 
subtended  by  the  semi-di- 
ameter of  the  earth's  orbit. 
Observations  of  Jupiter  taken 
from  diametrically  opposite 
points  of  the  earth's  orbit 
give  the  annual  parallax  of 
Jupiter  as  11°  4'  58".  Find 
the  distance  of  the  sun  from 
Jupiter. 

6.  The  annual  parallax  of 
a  Centauri,  the  nearest  fixed 
star,  is  0.916".  Find  the  dis- 
tance of  the  star  from  the 
sun.  (Note  that  for  ver 
small  angles  tan  B  may 
replaced  by  9>  Use  the  vah 
of  1"  in  radians  found  at  tl 
bottom  of  the  front  cover 
the  Tables.)  If  light  trave 
at  the  rate  of  186,320  mil€ 

per  second,  find  the  length  of  time  necessary  for  light  to  traverse  tl 

distance. 
7.  The  annual  parallax  of  the  North  Star  is  0.073".    Find  the  distance 

to  the  North  Star  in  light-years. 

*  Inspection  of  the  Condensed  Table  on  the  inside  of  the  back  cover  of 

the  Tables  shows  that  values  of  Q,  in  radians,  and  of  tan  Q  agree  to  three 

decimal  places  if  ^  =  6°;  to  four  places  if  0  =  3°. 


Fig.  138. 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    237 

8.  Find  the  time  that  it  takes  light  to  reach  the  earth  from  the  sun; 
to  reach  Neptune,  the  most  remote  planet,  distance  2,788,000,000  miles. 

B?  86.  Exponential  Equations.  An  equation  in  which  the  un- 
known enters  as  an  exponent  is  called  an  exponential  equation. 
Exponential  equations  of  the  type  a^^^^  =  6^^*^,  where  f{x) 
and  F{x)  are  hnear  or  quadratic  algebraic  functions  may  be 
solved  by  equating  the  logarithms  of  the  two  sides  of  the  equa- 
tion and  solving  the  resulting  algebraic  equation,  as  in  the  fol- 
lowing examples. 

Example  1.     Find  the  function  of  the  type  y  =  x"  whose  graph  passes 
through  the  point  (3,  5). 
Substituting  the  coordinates  of  the  point  in  the  equation,  we  have 

S**  =  5. 
Equating  the  logarithms  of  both  sides  of  this  equation,  we  have 

log  3"  =  log  5  J 

or  n  log  3   =  log  5. 

Hence  0.4771n  =  0.6990 

_  0.6990       log  0. 6990  =  0. 8445  -  I 
^  ~  0.4771       log  0.4771  =0.6786  -1 
=  1.4653  log  n  =  0.1659 

Hence  the  required  function  is 

y  =  xi-<653. 

The  value  of  n  is  here  calculated  from  the  quotient  by  logarithms  rather 
than  by  long  division.  Notice  that  if  0.4771  were  subtracted  from  0.6990, 
the  result  would  be  log  (5/3)  and  not  log  5/log  3. 

Example  2.  Find  the  logarithm  to  the  base  e  of  any  positive  number 
X  assuming  a  table  of  common  logarithms  to  be  given.     Let 

y  =  log«  X,  hence  e"  =  x. 

Equating  the  common  logarithms  of  both  sides  of  the  second  equation 
we  have 

logio  e"  =  logio  X,     whence        y  logio  e  =  logw  x. 

Therefore  y  =  , logw  x, 

logio  e 

Since  logw  e  =  0.4343,  we  have  log,  x  =     ^^^   logio  x.  (I) 

That  is,  the  logarithm  to  the  base  e  of  any  number  can  be  found  by  multi- 
plying the  common  logarithm  of  the  number  by  1/0.4343. 


238 


ELEMENTARY  FUNCTIONS 


For  instance  we  have 
log.  6.75  = 
1 


logio6.75 


0.8293 


1.91. 


y 

—3 

^ 

lo 

1i 

flfeX 

F 

L 

_ 

^ 

^ 

Qio 

xsE 

2.3( 

m 

^ 

■^ 

fl' 

x~ 

O 

y 

T  i 
0.4 

e 

343. 

^     '■ 

I 

) 

s 

f 

0.4343  ^'""     0.4343 
The  constant  ^  ^^.^  is  called  the  modulus  of  the  base  e  with  respect  to 

the  base  10. 

The  figure  shows  the  graphs  of  logio  x  and  log«  x. 

If       x  =  OP  J        then        logio  x  =  PQ,        and        log,  x  =  PR. 

If  OA  =  e,        then        AB  =  logio  e. 

Hence  equation  (1)  may  be  written  in  the  form  PR  =  -~  =  — - —    PQ 

AH         \j .  4:o4o 

and  the  graph  of  log*  x  can  be  obtained 
from  that  of  logio  x  by  the  theorem  on 
page  89. 

Example  3.     The  variation  of  at- 
mospheric pressure  p  (in  millimeters  of 
mercury  in  a  barometer)  with  respect 
to  the  altitude  h  above  sea  level  (in 
Fig.  139.  meters)    is    given    by    the    equation 

p  =  jhe'^,  where  po  and  m  are  con- 
stants and  e  is  the  base  of  the  natural  system  of  logarithms.  If  p  =  719 
when  h  =  450,  and  p  =  594  when  h  =  2000,  find  p  U  h  =  8000. 

Before  starting  the  solution,  notice  that  po  is  the  atmospheric  pressure 
at  sea  level;   for  ii  h  =  0,  p  =  poe~^  =  poe°  =  po. 

A  more  convenient  form  of  the  given  equation,  for  purposes  of  compu- 
tation, is  obtained  by  equating  the  common  logarithms  of  both  sides  of 
the  equation,  which  gives: 

log  p  =  log  Po  -  mh  log  e.  (1) 

Substituting  the  given  data  we  have  the  two  equations 

log  719  =  log  Po  -  450  X  0.4343  m,  (2) 

log  594  =  log  Po  -  2000  X  0.4343  m.  (3) 

These  equations  may  be  solved  simultaneously  for  the  constants  m  and 
poi  as  follows: 
Subtracting 
log  719  -  log  594  =  1550  x  0. 4343w 

2.8567-2.7738  0.0829 


Hence  m 


log  1550  =  3. 1903 
log  0.4343  =  0.6378-1 


1550x0.4343 
wi  =  0.0001232 


1550x0.4343 


log  0.0829 
log  m 


2.8281 
0.9186 


0.0905-4 


Substituting  this  value  of  m  in  (2)  we  have 
log  Po  =  2. 8567  4-  450  x  0. 4343  X  .  0001232 

=  2. 8567  4-  0. 02407,  

log  Po  =  2. 8808  log  450  x  0. 4343  m  =  0. 3815  -  2 

Po  =  760.  450  X  0. 4343  m  =  0. 02407 


log  450  =  2.6532 
log  0.4343  =  0.6378-1 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    239 

Hence  p  =  7Q0e-^-^^^'^'^. 

Uh  =  8000,    p  =  760e-o-985«,  log  0. 9856  =  0. 9937  -  1 

i      log  p  =  log  760-0. 9856  log  e.  log  0. 4343  =  0.6379  -I 

r               =2.8808-0.4282  0.6316-1 

=  2. 4526.  0. 9856  log  e  =  0. 4282 
Hence  p  =  283.5  if /i  =  8000. 


EXERCISES 

1.  Solve  for  x, 

(a)  7*  =  4.  (b)  4*-i  =  9.  (c)  4^  =  7.  (d)  S^'-^  =  5*"'. 

2.  Each  of  the  persons  receiving  one  of  the  letters  mentioned  in  Exercise 
18,  page  232  was  requested  to  forward  10  cents.  Find  the  number  on  the 
letters  at  which  the  chain  should  have  been  completed  if  the  object  had 
been  to  raise  $1,000,000. 

3.  The  population  of  the  United  States  in  1898  was  approximately 
75,000,000.  Find  the  number  on  the  letter  at  which  the  chain  in  Exercise 
18,  page  232,  should  have  been  completed  if  it  was  expected  that  every 
man,  woman,  and  child  in  the  country  would  receive  one. 

4.  Would  the  method  of  Section  85  enable  one  to  solve  the  equation 
2*  +  3*  =  7? 

5.  Find  a  function  of  the  type  y  =  z^  which  passes  through  the  point 
(3,  2).  Find  the  average  rate  of  change  of  this  function  from  x  =  3  to 
a;  =  4. 

6.  Find  the  common  logarithm  of  any  positive  number  x,  assuming 
that  a  table  of  logarithms  to  the  base  e  is  given. 

7.  Find  the  numerical  value  of  M  =  logio  e  (e  =  2.718)  from  the  table 
of  common  logarithms.  By  the  method  df  Example  2,  Section  85,  find  the 
value  of  m  =  loge  10.     Find  the  reciprocal  of  M,  and  compare  it  with  m. 

8.  Assuming  that  a  table  of  logarithms  to  the  base  h  is  given,  show 
that  the  logarithm  of  any  positive  number  x  to  the  base  a  is 

,  logt  X 

l0gaX  =  ,-^ 

logi  a 

Hence  show  that  log*  10  = , ,  or  log«  10  logio  e  =  1. 

logio  e 

9.  By  means  of  the  preceding  exercise  show  that 

loge  X  =  m  logio  X,         where         m  -2. 3026, 
and  logio  x  =  JVf  loge  X,         where        M  =  0.4343. 

i  Illustrate  these  two  equations  on  the  graphs  of  loge  x  and  logw  x. 

Note:  The  system  of  common  logarithms,  in  which  the  base  is  10, 
and  that  of  natural  logarithms,  in  which  the  base  is  e  =  2.718,  are  the  most 
important  systems  of  logarithms.    The  former  is  used  for  numerical 


240  ELEMENTARY  FUNCTIONS 

computations,  the  latter  for  most  theoretical  work.  The  relations  be^ 
tween  these  two  systems  obtained  in  Exercises  6  to  9  may  be  summed  up 
as  follows: 

If  the  common  logarithm  of  a  number  is  given,  the  natural  logarithm  is 
found  by  multiplying  it  by 

m  =  ,— i-  =logJ0  =  2.3026. 
logix)  e 

The  constant  m  is  called  the  modulus  for  changing  from  the  common  to 
the  natural  system. 

If  the  natural  logarithm  of  a  number  is  given,  the  common  logarithm  is 
found  by  multiplying  it  by 

which  is  called  the  modulus  for  changing  from  the  natural  to  the  common 
system. 

10.  Solve  Example  3,  Section  85,  using  the  table  of  natural  logarithms 
(page  31  of  the  Tables). 

11.  Find  log2  3,  log2  5,  log2  7,  and  then  by  means  of  some  of  the  funda- 
mental properties  of  logarithms  (Section  81,  properties  7,  8,  9)  construct 
a  table  of  logarithms  to  the  base  2  for  the  integers  1  to  10  inclusive. 

12.  A  hawser  of  a  ship  is  subjected  to  a  strain  of  6  tons.  How  many 
turns  must  be  taken  around  a  post  in  order  that  a  man  who  can  not  pull 
more  than  200  pounds  may  keep  the  hawser  from  slipping,  if  the  coeflBcient 
of  friction  m  =  0. 175  and  S  =  Pe"^,  where  S  =•  strain  in  the  hawser  in 
pounds,  P  =  pull  of  the  man  in  pounds,  9  =  the  angle  of  contact  of  the 
hawser  and  post  in  radians. 

13.  A  steamer  approaching  a  dock  has  a  velocity  of  20  feet  per  second 
at  the  instant  the  power  is  shut  off,  and  10  feet  per  second  at  the  end  of 
1  minute.  Find  the  velocity  at  the  end  of  2  minutes  if  the  law  of  motion 
is  y  =  Voe~*^K 

14.  In  a  chemical  experiment  it  was  found  that  at  the  end  of  1  hour 
there  were  35.6  c.c.  of  a  substance  in  solution  and  at  the  end  of  3  hours, 
18.5  c.c.  If  the  amount  A  of  the  substance  at  any  time  t  is  given  by  the 
equation  A  =  /ce"^',  determine  the  time  that  elapsed  before  the  amount 
was  reduced  to  5  c.c. 

16.  The  intensity  of  light  /  after  passing  through  a  medium  of  thick- 
ness T  is  7  =  7oe~^r.  If  light  loses  3  %  of  its  intensity  in  passing  through 
a  lens,  what  per  cent  of  intensity  will  remain  after  passing  through  four 
lenses? 

16.  Radium  decomposes  so  that  the  amount  A  remaining  after  a  time 
t  is  A  =  Aoe~^.  If  I  %  disappears  in  20  years,  how  long  before  one-half 
of  the  original  amount  will  be  gone? 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     241 

17.  As  a  body  cools,  at  any  time  t  the  difference  in  temperature  $  be- 
tween the  body  and  the  surrounding  medium  is  given  by  the  equation 
6  =  doe~"^.  If  a  fireless  cooker  oven  is  430*'  above  room  temperature 
when  t  =  0,  and  216**  after  one  hour,  how  long  before  it  will  be  10°  above 
room  temperature? 

86.  Compound  Interest.  Interest  is  the  money  paid  for  the 
use  of  borrowed  capital  which  is  called  the  principal.  Interest 
that  is  paid  only  on  the  principal  is  called  simple  interest. 
When  interest  is  paid  on  unpaid  interest,  which  is  added  to  the 
principal  periodically,  it  is  called  compound  interest.  If  un- 
paid interest  is  added  to  the  capital  at  yearly  intervals  it  is 
said  to  be  compounded  annually.  If  the  interest  is  added  every 
six  months  or  three  months  it  is  said  to  be  compounded  semi- 
annually or  quarterly. 

In  any  problem  in  compound  interest  where  the  interest  is 
compounded  annually,  four  quantities  are  involved: 

P,  the  principal ; 

i,  the  rate  of  interest,  which  is  the  sum  of  money  in  dollars 
paid  for  the  use  of  one  dollar  for  one  year.  Thus,  if  the  rate 
is4%,i  =  $0.04; 

n,  the  number  of  years; 

S,  the  amount,  which  is  the  sum  of  the  principal  and  interest 
for  n  years. 

Theorem  1.     The  amount,  S,  of  a  principal  of  P  dollars j  in- 
terest compounded  annually  for  n  years  at  the  rate  ^,  is 
S  =  P(l  +  i)\ 

The  amount  of  one  dollar  for  one  year  at  the  rate  t  is  1  +  i. 

Hence  the  amount  of  P  dollars  for  one  year  is  P(l  +  i). 

The  amount  at  the  end  of  the  second  year  is  obtained  by 
multiplying  the  new  principal,  P(l  +  i)>  by  the  amount  of  one 
dollar  for  one  year  1  + 1. 

Hence   the    amount    at   the   end   of   the   second   year   is 

p(i  +  i)(i  +  i)  =p(i  +  iy. 

Similarly,  the  amount  at  the  end  of  the  third  year  is 
P(l  +  iy{l  +  i)  =  P(l  +  iy.  Comparing  these  results,  we  see 
that  the  exponent  of  (1  +  i)  is  the  same  as  the  number  of 
years,  and  hence 


242  ELEMENTARY  FUNCTIONS 

The  amount  S  at  the  end  of  n  years  is  >S  =  P(l  +  i)". 

This  equation  involves  four  variables.  If  any  three  are 
given,  the  fourth  may  be  determined  by  substituting  the  given 
values  and  solving  for  the  unknown.  The  value  of  S,  P,  or  i 
may  be  determined  by  algebraic  processes  while  the  determina- 
tion of  n  involves  the  solution  of  an  exponential  equation. 

Example  1.    At  what  rate  will  $50  amount  to  $75  in  7  years? 
Here    P  =  50,     *S  =  75,     and    n  -  7. 

Hence  75  =  50  (1  +  iy,  log  1 . 5  =  0. 1761. 

Then  (1  +  i)7  =  ^ 5_     ^15^  ^  log  1 . 5  =  0. 0252. 

So  that        (1  +  i)  =  v^O  v^Es  =  1 .  0598. 

=  1.0598, 
Hence  i  =  0. 0598,  so  that  the  rate  is  5. 98  %. 

If  the  interest  is  compounded  semi-annually  (as  in  some 
savings  banks),  in  n  years  there  are  2n  periods  and  the  interest 
on  one  dollar  for  each  period  is  i/2,  so  that  the  amount  in  n 
years  is 

If  the  interest  is  compounded  quarterly,  the  amount  in  n 
years  is 

In  general,  we  have 

Theorem  2.  If  interest  is  compounded  m  times  a  year,  at 
the  annual  rate  i,  the  amount  after  n  years  is  the  same  as  if  the 
interest  were  compounded  annually  at  the  rate  i  /m,  for  mn  years. 
That  is 

S  =  P(1  +i7m)'"\ 

Example  2.  Find  the  amount  of  $50  for  10  years  with  interest  con- 
vertible into  principal  semi-annually  at  4  %. 

Here  P  =  50,  i  =  .  04,  n  =  10,  m  =  2,  i/m  =  .  02  and  mn  =  20. 
I    Hence  ^1  =  50  (1  +  .  02)2o  log  50  =  1 .  5690 

=  50  (1.02)20  20  log  1.02  =  0. 1720 

=  $74.30.  log  5  =  1.8710 

Definition.  The  present  value,  P,  of  a  sum  of  money,  S, 
due  after  n  years,  is  the  principal  which  must  be  placed  at 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    243 

compound  interest  at  a  given  rate  i,  in  order  to  amount  to  the 
sum  S  at  the  end  of  n  years. 
Thus  in  the  formula  in  Theorem  1,  P  is  the  present  value  of  S. 

EXERCISES 

1.  A  principal  of  $100,  deposited  in  a  trust  company,  bears  interest  at 
the  rate  of  4%  compounded  semi-annually.  What  will  the  balance  be 
at  the  end  of  ten  years  if  no  withdrawals  are  made? 

2.  A  mother  promises  her  twelve  year  old  boy  that  she  will  present  him 
with  $100  on  his  twenty-first  birthday  provided  he  abstains  from  smok- 
ing until  that  time.  How  much  should  she  deposit  in  the  trust  company 
of  the  previous  example  in  order  to  have  an  amount  of  $100  when  the  boy 
is  twenty-one?  ^      ■'       ^s^ 

3.  At  what  rate  would  a  siun  of  money  double  itself  in  25  years  if  inter- 
est is  compounded  annually? 

4.  In  how  many  years  will  a  sum  of  money  double  itself  if  placed  at  com- 
pound interest  if  (a)  the  rate  of  interest  is  5%,  compounded  annually? 
(b)  the  rate  is  4  %  compounded  semi-annually? 

5.  If  $100  be  deposited  in  a  trust  company  at  4  %,  compounded  semi- 
annually, how  long  before  it  will  amount  to  at  least  $150? 

6.  A  building  and  loan  association  offers  an  opportunity  for  an  invest- 
ment to  yield  8  %,  compounded  quarterly. 

(a)  If  $100  is  invested,  to  what  will  it  amount  in  5  years? 

(b)  What  sum  should  be  invested  to  amount  to  $120  in  7  years? 

(c)  How  long  must  $50  be  invested  to  amount  to  $175? 

»(d)  How  long  will  it  take  any  principal  to  double  itself? 
(e)  How  long  will  it  take  for  a  sum  to  treble  itself? 

7.  A  man  borrows  $50  from  a  friend,  and  three  years  later  returns  $65. 
What  is  the  equivalent  rate  of  interest,  if  interest  is  compounded  quarterly? 

8.  A  man  bought  a  diamond  for  $150  in  1900,  and  sold  it  for  $400  in 
1915.  What  rate  of  interest  did  he  realize,  assuming  it  to  be  compounded 
annually? 

9.  If  a  building  lot  is  bought  for  $500,  and  its  value  increases  by  10  % 
annually,  what  will  it  be  worth  in  5  years? 

10.  The  number  of  students  in  a  college  increased  from  275  to  460  in 
10  years.  At  what  rate  did  the  number  increase,  assuming  that  the  per- 
centage of  increase  each  year  was  constant? 

11.  What  sum  should  be  deposited  in  a  bank  paying  3  %  compounded 
semi-annually  in  order  to  pay  off  a  debt  of  $500  due  three  years  later? 

12.  Construct  the  gra,ph  of  the  amount  of  one  dollar,  interest  com- 
pounded annually  at  6  %,  as  a  function  of  the  number  of  years  n.  Build 
the  table  of  values  for  n  =  10,  20,  30,  40. 


244  ELEMENTARY  FUNCTIONS 

13.  Verity  the  fact  that  the  values  of  S  obtained  in  Exercise  12  are  In 
geometrical  progression. 

87.  Annuities.  Definition.  An  annuity  consists  of  a  series 
of  equal  payments  made  at  equal  intervals  of  time.  The  first 
payment  of  an  annuity  is  made  at  the  end,  not  at  the  beginning, 
of  the  first  period  of  time,  unless  otherwise  specified. 

Annuities  are  common  in  commercial  life.  The  rent  of  a 
house  or  store,  the  premium  of  an  insurance  policy,  the  dividend 
on  a  bond,  wages  and  salaries,  the  payments  of  interest  on  a 
mortgage,  are  examples  of  annuities. 

Theorem  1.  The  amount  of  an  annuity  of  R  dollars,  payable 
annually,  accumulated  for  n  years  at  the  rate  i,  interest  compounded 
annually  is, 

1 

The  compound  interest  on  each  payment  is  obtained  by  the 
method  of  the  preceding  section. 

Since  the  first  payment  is  made  at  the  end  of  the  first  year, 
the  first  payment  is  at  interest  for  n  —  1  years  and  amounts 
to  R{1  +  i)^-K 

The  second  payment  is  at  interest  f or  w  —  2  years  and  amounts 
to  R{1  4-  ^>-^  etc. 

The  next  payment  to  the  last  is  at  interest  1  year  and  amounts 
to  R{1  +  i). 

The  last  payment  bears  no  interest  and  amounts  to  R. 

Adding  these  amounts  in  the  reverse  order  we  obtain 

K  =  R  +  R{l+i)+  .  .  .  -\-R{l+  ^>-2  +  R(l  +  i^-K 

The  terms  on  the  right  form  a  geometric  progression  whose 

sum  is  obtained  by  the  formula  S  =  — — r,  where  the  first  term 

18  a  =  R,  the  last  term  is  I  =  R(l  +  t)"~S  and  the  ratio  is 
r  =  1  +  i. 

Substituting,  we  have 

„      (l4^^)ig(l+^)"-^-/^       p  (1  +iy-l 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    245 

Theorem  2.  The  amount  of  an  annuity,  if  R  dollars  are  paid 
m  times  a  year,  accumulated  for  n  years  at  the  rate  i,  interest 
compounded  m  times  a  year,  is  equivalent  to  the  amount  of  an 
annuity  of  R  dollars,  payable  annually,  accumulated  for  mn 
years  at  the  rate  i  /m,  interest  compounded  annually. 

Definition.  The  present  value  of  an  annuity  of  R  dollars  a 
year  for  n  years  is  the  amount  that  must  be  deposited  in  a  bank, 
interest  compounded  annually  at  the  rate  i,  so  that,  if  R  dollars 
be  withdrawn  annually,  there  will  be  no  balance  left  after  the 
nth  withdrawal. 

Theorem  3.  The  present  valu£  of  an  annuity  of  R  cbllars 
payable  annually  for  n  years  is 

I 

The  present  value  of  the  annuity  is  the  sum  of  the  present 
values  of  R  dollars  due  1  year  hence,  R  dollars  due  2  years 
hence,  etc.,  for  n  years.  We  have  from  the  formula  for  com- 
pound interest  that  the  present  value  P  of  a  sum  S  due  in  n 
years  is  P  =  S{1+  t)~".  Hence  if  A  denotes  the  present  value 
of  the  annuity,  we  have 

A  =  P(l  +  i)-i  +  R{1  +  i)-^  +  •  •  •  -f  P(l  +  i)-\ 
The  sum  on  the  right  is  a  geometric  progression  for  which 
a  =  P(l  +  i)-\   ?  =  /2(1+  ^)-^   and    r  =  (1  +  i)-\      Substi- 
tuting these  values  in  the  formula  for  the  sum  of  a  geometric 
progression,  we  have 

,  _  (1  +  i)-'R{l  +  ^)-"  -  P(l  +  i)-' 
(1  +  i)-i  -  1 

'Multipl3dng  the  numerator  and  denominator  of  the  right-hand 
side  by  (1  +  i)+i  and  simpUfying,  we  have 

1  -  (1  + 1)  I 

Theorem  4.     The  present  value  of  an  annuity,  if  R  dollars 
fare  paid  m  times  a  year  for  n  years,  at  the  rate  i  compounded 
m  times  a  year,  is  equivalent  to  that  of  an  annual  annuity  of  R 
dollars  for  mn  years  at  the  rate  i/m  compounded  annvully. 


246  ELEMENTARY  FUNCTIONS 

EXERCISES 

1.  If  $50  be  deposited  annually  in  a  building  and  loan  association  pay- 
ing 6%,  compounded  annually,  what  will  the  savings  amount  to  in  10 
years? 

2.  How  much  must  a  man  save  annually,  and  deposit  in  a  savings  and 
loan  company  paying  5%,  compounded  annually,  in  order  to  pay  off  a 
mortgage  for  $2000  after  5  years? 

3.  How  long  will  it  take  to  accumulate  $2000  if  $75  are  deposited  an- 
nually in  a  savings  and  loan  company  paying  6  %,  compounded  annually? 

4.  A  father  buys  a  bond  for  $1000,  due  after  18  years,  which  bears 
interest  at  5  %  payable  semi-annually,  for  the  college  expenses  of  his  in- 
fant boy.     The  interest  payments  are  deposited  in  a  bank  paying  4% 
compounded   semi-annually.     How   much   money   will   be   available    IJ 
years  later? 

5.  A  man  buys  a  house  and  lot,  paying  $1000  down,  and  agreeing 
pay  $1000  annually  for  4  years.  What  is  the  equivalent  cash  price 
money  is  worth  6  %  per  annum?  Note  that  the  first  payment  is  not  p 
of  the  annuity,  since  the  first  payment  of  an  annuity  is  due  at  the  end 
the  first  period. 

6.  A  piano  is  sold  for  $100.'cash  and  $25  quarterly  for  2  years.  What  is 
the  equivalent  cash  price,  if  money  is  worth  4%  compounded  quarterly? 

7.  A  man  buys  a  house,  giving  back  a  mortgage  for  $5000  with  interest 
payable  annually  at  6%.  If  the  mortgage  is  to  be  paid  off  by  five  equal 
annual  payments,  covering  principal  and  interest,  what  should  be  the 
amount  of  the  annual  payment?  Hint:  The  annual  payments  consti- 
tute an  annuity  whose  present  value  is  $5000. 

8.  Which  is  the  better  offer  for  a  house  from  the  seller's  standpoint, 
$5000  down  and  $1000  annually  for  4  years,  or  $5000  down  and  $2000 
annually  for  2  years,  if  money  is  worth  4  %? 

9.  Six  months  before  a  boy  enters  college,  his  father  wishes  to  deposit 
a  sum  in  a  savings  bank  paying  4  %,  interest  compounded  semi-annually, 
which  will  enable  the  boy  to  draw  $300  every  6  months  during  his  college 
com'se.     How  much  should  he  deposit? 

10.  It  is  estimated  that  a  certain  mine  will  be  exhausted  in  10  years. 
If  the  mine  yields  a  net  annual  income  of  $2000,  what  would  be  a  fail 
purchase  price,  money  being  worth  5  %? 

11.  A  man  holds  a  mortgage  for  $6000,  due  after  3  years,  interest  6  % 
payable  quarterly.  If  he  disposed  of  it  how  much  might  he  expect  to  rei- 
ceive,  if  money  is  worth  4  %  compounded  quarterly  .  Hint:  The  present 
value  of  the  mortgage  is  the  sum  of  the  present  value  of  $6000  due  after  3 
years  and  of  a  quarterly  annuity  of  $90  (one  fourth  of  a  year's  interest) . 

12.  A  bond  for  $100  is  to  be  redeemed  at  the  end  of  10  years,  and  bears 
interest  at  5  per  cent,  payable  semi-annually.    At  what  price  for  the  bondj 


i 

^    EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     247 

will  the  purchaser  realize  4  per  cent  on  his  investment?  Hint:  The  present 
value  of  the  bond  consists  of  the  present  value  of  $100  due  10  years  hence 
and  the  present  value  of  the  annuity  consisting  of  the  semi-annual  in- 
terest payments. 

13.  What  is  the  purchase  price  of  the  bond  in  the  preceding  exercise 
if  the  buyer  is  to  reahze  6  per  cent? 

14.  An  insurance  company  desires  to  offer,  at  the  time  a  policy  matures, 
an  option  between  a  20-year  bond  for  $1000,  bearing  interest  at  5  per  cent, 
payable  semi-annually,  and  an  equivalent  cash  payment.  If  the  computa- 
tions of  the  company  are  on  a  3  per  cent  basis,  what  should  the  cash  pay- 
ment be? 

15.  At  the  maturity  of  an  insurance  policy,  the  company  offers  a  cash 
payment  of  $1000  or  the  option  of  five  equal  annual  payments,  the  first 
to  be  made  one  year  after  the  policy  matures.  If  the  company  assumes 
that  money  is  worth  3  per  cent,  what  should  be  the  amount  of  the  annual 
payment? 

16.  Three  months  before  a  boy  enters  college,  his  father  deposits  on 
the  boy's  account  $1200  in  a  bank  paying  4  per  cent  interest,  compounded 
quarterly.  The  boy  wishes  to  withdraw  the  money  quarterly,  in  equal 
amounts,  during  the  four  years  of  his  course.  How  much  should  he  draw 
at  a  time? 

17.  A  city  issues  20-year  bonds  to  the  amount  of  $100,000  in  order  to 
raise  money  for  the  improvement  of  its  water  supply.  A  sinking  fund  to 
provide  for  the  extinction  of  the  debt  can  be  accumulated  at  4  per  cent, 
interest  compounded  annually.  How  much  must  be  deposited  in  the 
sinking  fund  at  the  end  of  each  year? 

18.  A  man  buys  an  automobile  for  $1000,  and  estimates  that  he  will 
be  allowed  $400  for  it  in  purchasing  a  new  car  three  years  later.  How 
much  should  he  save  every  three  months  for  the  purchase  of  the  new  car, 
if  he  deposits  his  savings  in  a  bank  paying  4  per  cent  interest,  compounded 

•  quarterly? 

19.  A  furnace  costing  $450  must  be  installed  in  a  house  every  10  years. 

*  How  much  should  the  landlord  save  each  year  for  this  purpose,  if  the  sav- 
ings can  be  accumulated  at  5  per  cent,  interest  compounded  annually? 

20.  A  man  owes  $2000  on  which  he  pays  5  %  interest.  If  he  pays  the 
debt,   principal   and  interest,  in   6   equal  annual  installments,  what  is 

,    the  amount  of  each  payment?     How  much  will  he  owe  at  the  beginning  of 

the  second  year?    At  the  beginning  of  the  third  year? 
I      21.  Find  the  Hmit  of  the  present  value  of  an  annuity  as  the  number  of 
f  payments  increases  indefinitely.     Verify  the  result  by  finding  the  sum 
?   of  the  infinite  geometric  progression  whose  terms  are  the  present  values  of 
the  several  payments. 

22.  A  college  graduate  wishes  to  provide  for  a  scholarship  of  $90  a 
year.     Find  the  amount  which  he  must  present  to  the  college,  if  collegiate 


248 


ELEMENTARY  FUNCTIONS 


--f/f 

"vir 

wM 

jpi^  '  ^ 

J-S^L 

--- — \- 

'      r 

funds  can  be  invested  at  4^  %,  using  the  result  obtained  in  the  preceding 
exercise.     By  what  other  method  can  the  amount  be  found? 

23.  A  raih-oad  spends  S900  annually  to  provide  a  watchman  for  a  grade 
crossing.  If  money  is  worth  5  per  cent,  how  much  can  they  reasonably 
be  expected  to  spend  for  the  elimination  of  the  crossing? 

88.  Graph  of  the  Exponential  Function  kh""" .  The  form  of 
the  graph  of  If  was  determined  on  page  216.  From  it  the  graph 
of  Mf  may  be  found  by  multiplying  ordi- 
nates  by  k  (Theorem,  page  89).  Fig.  140 
gives  the  graph  of  /c2^  for  several  values  of  fc. 
Notice  that  i/  =  A;  is  the  intercept  on  the 
2/-axis. 

The  form  of  the  graph  of  the  function  6"* 
may  be  found  by  dividing  by  n  the  abscissas 
of  several  points 
on  the  graph  of 
¥  (Theorem, 
page  151).  The 
graph  of  2"^  for 
several  values  of 
n  is  given  in  Fig.  141. 

The   combination  of    the   two 
theorems  quoted  shows  that  the 
graph  of  kb^  may  he  obtained  as 
follows:     Plot  the  graph  of  If,  divide  the  abscissas  of  points 
on  it  by  n,  which  gives  the  graph  of  6"^,  and  multiply  the  ordir 

rmtes  of  points  on  the  new   graph 
by  k. 

Example.  Sketch  the  graph  of  2  {2'"^). 
Here  k  =  2  and  n  =  \.  Dividing  the  ab- 
scissas by  \  amounts  to  multiplying  them 
by  3.  Hence  we  first  construct  the  graph 
of  2*  and  multiply  the  abscissas  of  points 
on  it  by  3.  The  ordinates  of  points  on 
the  curve  so  obtained  are  then  multi- 
plied by  fc  =  2. 

The  form  of  the  graph  of  A;6"*  depends  on  the  value  of  6,  but  for  suitable 
values  of  k'  and  n'  the  same  curve  is  the  graph  of  k'c^'*.    We  shall  use 


FiQ.  140. 


J/>              _[_  " 

7-        ?»/      7-           .Am.1             '■I'.L 

-ujtt    Mi    3^ 

I^JII     /^    7 

Mtt  y  7 

-'4/    2^7 

-IM^-^ 

^r 

-f-jo           2     ;      4          .'■■   |_    ^ 

Fig.  141. 


JL        _j         /        L 

■        ■fLj$g__ih' 

ty  / 

-J7  ^7 

7^  y 

^  ^ 

":^ 

T  ■     0      1     2     3     i     i     6     7     r  i 

FiQ.  142. 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    249 

b  =-  10  as  a  standard  value,  and  we  shall  now  show  that  we  can  determine 
a  constant  m  such  that  the  graphs  of  kb"^  and  A;  10"^  are  identical. 
The  graphs  will  be  identical  if  and  only  if 

or  b'^  =  10'^. 

This  is  essentially  an  exponential  equation  to  be  solved  for  m.  Equat- 
ing the  logarithms  of  both  sides  of  the  equation, 

nx  log  b  ='  mx  log  10  =  mx, 
whence  m  =  n  log  6.  •» 

Then  kb"^  =  fclO"  * '"«'  \ 

Hence  it  is  always  possible  to  use  the  base  10,  and  we  shall  do  so  in  the 
future  unless  the  contrary  is  indicated. 

The  most  characteristic  properties  of  the  graph  of  y  =  klO""" 
are  that  it  does  not  cross  the  x-axis,  that  it  does  cross  the^-axis, 
and  that  it  always  rises  or  always  falls. 


EXERCISES 

1.  Plot  the  graphs  of  the  functions: 

(a)  T^(22*).       (b)  K2^).         (c)  3(20-6').         (d)  |(2o-»*).        (e)  2(3'^*). 
(f)   3(2-0-25*).  (g)  2(3-0-^).  (h)  0.25(2^).        (i)    0.5(10^). 

2.  Plot  the  graph  oi  y  =  e~^.     How  can  the  graph  ol  y  =  ke~  ^  be 
obtained  from  it?     This  graph  is  called  the  probability  curve. 

3.  Find  the  inverse  of  the  function  A:10^.     What  are  the  most   im- 
portant characteristics  of  the  graph  of  i/  =  c  log  nx? 

4.  (a)  Construct  the  graph  of  k  log2  x  for  several  values  of  k. 

(b)  Construct  the  graph  of  log2  nx  for  several  values  of  n. 

(c)  Construct  the  graph  of  ^  log2  3x. 

89.  The  Logarithmic  Scale.    The  ordinates  of  points  on  the 
raph  of  logio  x  corresponding  to  integral  values  of  x  from  1  to  10 


2/ 
A 

- ..  _^_,.i^.^      ij 

~      "" 

1 

^^^    T 

0 

12         3         4 

» 

)          I 

1.          ..          .          ^ 

0    X 

Fig.  143. 


inclusive,  are  a  set  of  segments  on  the  2/-axis  whose  lengths  from 
the  origin  are  equal  to  the  logarithms  of  the  corresponding 
abscissas. 


250  ELEMENTARY  FUNCTIONS 

The  line  O'A'  is  an  enlargement  of  the  segment  OA  on  the 
y-axis,  the  numbers  on  the  right  being  the  logarithms  of  those 

on  the  left. 
10^1.00         A  scale  is  called  a  uniform  scale  if  the  distance  of 
a  number  from  the  point  marked  0  is  equal  to  the 
^^      number,  the  distance  from  0  to  1  being  the  unit 
segment. 

A  scale  is  called  a  logarithmic  scale  if  the  distance  of 
a  number  from  the  point  marked  1  is  equal  to  the 
logarithm  of  the  number,  the  distance  from  1  to  10 
being  taken  as  unity. 

The  logarithms  are  spaced  uniformly  along  the 
Hne  O'A'j  while  the  integers  are  spaced  non-uni- 
formly. 

The  utility  of  a  logarithmic  scale  lies  in  the  fact 
that  the  addition  and  subtraction  of  the  logarithms 
of  nimibers,  and  hence  the  multiplication  and  division 
of  numbers,  may  be  effected  mechanically  by  the 
addition  and  subtraction  of  line  segments  on  the 
Fig  144  scale.  For  instance,  to  find  the  product  3x2,  we 
add  the  segment  1  -  2  to  the  segment  1-3.  For 
this  purpose,  place  a  pair  of  dividers  so  that  the  points  are 
on  the  extremities  of  the  segment  1-2,  and  then  with  this 
opening  place  one  end  of  the  dividers  on  the  point  3  and  the 
other  will  touch  the  scale  exterior  to  the  segment  1  -  3  in 
the  point  6,  the  required  product. 

To  find  the  quotient  |,  subtract  the  segment  1-2  from 
the  segment  1—3  in  a  similar  manner  by  means  of  the 
dividers. 

The  logarithmic  scale  is  sometimes  called  Gunter's  scale 
after  Edmund  Gunter,  who  first  made  use  of  it  for  purposes  of 
calculation  in  1620. 

The  dividers  may  be  dispensed  with  if  two  identical  logarith- 
mic scales  are  arranged  to  slide  along  one  another  as  shown  in 
Fig.  145,  which  shows  the  method  of  finding  the  product 
3x2  and  the  quotient  3/2. 

For  the  product  4x5,  the  above  method  gives  a  point  out- 


o' 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     251 

side  the  scale.  To  avoid  this  we  divide  one  of  the  factors  by 
10,  then  multiply  by  the  other  factor  and  move  the  decimal 
point  one  place  to  the  right  in  the  result. 


7     8     910 


910 


4  5 


7     8    910 


i  5        6      7     8     910 


3x2  =  6. 


3  -^  2  =  1.5. 


Fig.  145. 


To  find  the  quotient  %,  divide  10  by  5,  multiply  by  4  and 
move  the  decimal  point  one  place  to  the  left  in  the  result.  The 
following  figures  illustrate  the  methods: 


4  S  S         7       8      9 1 


4  S  6         7       8      9  10 


4  S  6         7        8      9  1 


5  6         7       8      9  10 


4  X  5  =  (A  X  4)10  =  20.         I  =  (V-  X  4)tV  =  0.8. 

FiQ.  146. 


The  logarithmic  scale  finds  practical  application  in  the 
slide  rule,  an  instrument  much  used  by  engineers  in  calcu- 
lations. 

The  slide  rule  consists  of  a  ruler  with  a  central  portion  which 
slides  back  and  forth  in  grooves  on  the  rule,  the  slide  and  rule 


252  ELEMENTARY  FUNCTIONS 

being  graduated  with  equal  logarithmic  scales  which  are  labeled 
with  the  corresponding  numbers. 

The  ordinary  sHde  rule  has  four  scales  A,  B,  C,  D,  as  shown 
in  the  figure. 

Each  of  the  scales  C  and  Z)  is  a  single  logarithmic  scale. 
Each  of  the  scales  A  and  B  consists  of  two  logarithmic  scales, 
which  are  equal. 

If  we  give  to  the  left  index  on  all  four  scales  the  value  1, 
then  the  scales  A  and  B  extend  from  1  to  100  while  scales 


^    ^MmMnifi.Uii..ii!.ui..fi./.|^,r,|1lV 

^      1  I*  Is        %.      5    6    7  S  9  'l 


£""""'1^' 


i^fmmimA^^  f\  [^ 


i      5     6    7  8  91 


TTT 


FiQ.  147. 

C  and  D  extend  from  1  to  10.  Hence  the  scales  A  and  B, 
numerically  considered,  are  twice  as  long  as  scales  C  and  D, 
and  a  number  on  the  upper  scales  is  the  square  of  the  number 
below  it  on  the  lower  scales.  Conversely,  a  number  on  the  lower 
scales  is  the  square  root  of  the  number  directly  above  it  on  the 
upper  scales. 

The  methods  employed  to  keep  the  result  on  the  scales  in 
calculations  of  the  type  4x5  and  4  /5  are  not  necessary  if 
the  upper  scales  are  used.  The  lower  scales  give  more  accurate 
results,  since  the  graduations  are  not  so  fine  as  those  of  the 
upper  scales. 

The  instrument  is  provided  with  a  runner,  R,  which  enables 
coinciding  points  to  be  found  on  any  of  the  scales,  and  alsc 
permits  of  extensive  calculations  being  worked  out  without  the 
necessity  of  recording  the  intermediate  results. 

The  slide  rule  makes  use  of  matissas  only,  the  characteristic^ 
being  determined  by  inspection.  j 

Logarithmic  scales  are  also  used  on  the  axes  of  cross-sectioij 
paper.  If  the  scales  on  both  axes  are  logarithmic  the  cross| 
section  paper  is  called  logarithmic  paper.  If  the  scale  on  on^ 
of  the  axes  is  logarithmic  and  on  the  other  is  uniform  it  is  call«j 
semi-logarithmic  paper.  The  methods  of  using  these  cross 
section  papers  are  shown  in  the  following  examples. 


I 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    253 


Example  1.     Plot  the  graph  of  the  equation  obtained  by  taking  loga- 
rithms of  both  sides  of  the  equation  y  =  3x^. 

The  required  equation  is  log  ?/  =  log  3  +  2  log  x. 

Letting  log  y  =  Y  and  log  x  =  X,  we  have  Y  =  2X  +  log  3,  which  is  a 
linear  equation,  and  hence  its  graph  is  the  straight  line  whose  slope  is  2 
and  whose  intercept  on  the  F-axis 
is  log  3.  The  tables  give  corre- 
sponding pairs  of  values  of  x,  y  and 
of  X,  Y,  and  the  figures  show  cor- 
responding parts  of  the  graphs. 
Only  the  part  of  the  graph  of 
y  =  Sx^  in  the  first  quadrant  corre- 
sponds to  the  graph  in  the  X,  Y 
system,  as  the  logarithm  of  a  nega- 
tive number  is  not  a  real  number. 


y 


0 

0 

0.1 

.03 

1 

3 

2 

12 

0 

300 

X 

Y 

—  00 

—     00 

-1 

0.4771  - 

-2 

0 

0.4771 

0.3010 

1.0792 

1 

2.4771 

If  logarithmic  scales  are  con- 
tructed  on  the  X  and  F-axes  the 
graph  may  be  plotted  from  the 
table  of  values  of  x  and  y  since  the 
logarithmic  paper  serves  the  pur- 
pose of  finding  the  logarithms  as  is 
shown  in  the  following  example. 

Example  2.  Plot  the  graph  of 
the  equation  y  =  Sx^  on  logarithmic 
paper. 

Let  X  and  y  represent  the  numbers 
on  the  logarithmic  scales  on  the 
axes,  the  unit  lengths  1  to  10  being 
the  same,  and  X  and  Y  the  num- 
bers on  the  uniform  scales  attached  to  the  figure,  so  that  X  =  log  x  and 
Y  =  log  y. 

We  follow  the  same  procedure  as  in  plotting  the  graph  of  the  equation 
on  ordinary  cross-section  paper,  except  that  negative  values  of  x  and  y 
are  not  used,  and  the  lines  through  the  point  (1,  1)  are  usually  chosen  as 
the  axes  since  log  1  =  0. 


254 


ELEMENTARY  FUNCTIONS 


Y 

y 

12 

•10 
9 
8 
7 
6 
5 
4 

3 

i 

1 

1- 

/ 

/ 

/ 

/ 

1 

1 

/ 

/ 

0                 2            3        456789 10       OJ 

] 

L       jr 

3 
12 


The  point  (1,  3) 
will  be  at  the  inter- 
section of  the  lines 
a:  =  1  and  ?/  =  3. 
The  points  cor- 
responding to  other  pairs 
of  numbers  are  similarly- 
plotted.  These  points  lie 
on  the  straight  line  whose 
equation  Y  =  2X  +  log  3  was 
obtained  in  the  preceding 
example. 

Note  that  the  uniform 
scales  and  not  the  logarith- 
mic scales  determine  the 
slope.    Thus 


m  ■= 


log  12 


Fig.  149. 


log  2 
log22 
log  2 


-  log  3  _  log  4 

-  log  1  ~  log  2 
2  log  2 

log  2 


2. 


Theorem  1.     The  graph  of  the  power  function,  y=  fcx",  plotted 
on  logarithmic  paper  is 
the    straight    line    whose 
slope  is  n  and  whose  in- 
tercept on  the  y-axis  is  k. 

Let  X  and  Y  represent 
numbers  on  uniform 
scales  attached  to  the 
axes  so  that  X  =  log  x, 
and  Y  =  log  y. 

Taking  logarithms  of 
both  sides  of  the  equar 
tion  y  =  kx'^  we  have 
log  y  =  \ogk  +  n  log  x,  or 
Y  =  nX  -\-  Ky  the  graph 
of  which  referred  to  the 
uniform  scales  is  the 
straight  Une  with  slope  n 
and  intercept  K.    Hence, 


1 

i 

4 

:\ 

"1 

\ 

1  .  1 

\ 

\ 

.^ 

5^ 

\ 

A 

B      \ 

SI 

A^ 

7          \ 

yj 

^^ 

x/ 

7  ~  ^ 

^-^   X 

^  A% 

,    / 

c^ 

c\/ 

\ 

y 

2^-- 

\ 

» 

Cy/^ 

/ 

) 

\ 

0 

/ 

/ 

\ 
\ 

s 

O                 2            3        4       5    6    7 

8  9  10    a> 

" 

Fiq.  . 

L50. 

i  X 

EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    255 


using  the  notation  of  the  logarithmic  scales  in  place  of  X,  Y 
and  K,  we  have  the  theorem. 

Example  3.     Find  the  equation  of  the  line  marked  c  in  Fig.  150. 
The  intercept  on  the  t/-axis  is  2.    Since  the  line  passes  through  the 
points  (1,  2)  and  (10,  10)  the  slope  is 


log  10  -  log  2  _  log    5  _  0.6990 
log  10  -  log  1  "  log  10  ~       1 


=  0.6990. 


Hence  the  equation  is  y  =  2{lQP-^^'>). 

Example  4.  Plot  the  graph  oi  y  =  SCIO"-***)  on  semi-logarithmic  paper 
with  the  scale  on  the  x-axis  uniform  and  on  the  y-axis  logarithmic. 

Let  Y  represent  numbers  on  a  uniform  scale  attached  to  the  y-axis,  so 
that  Y  =  log  y. 

The  unit  hne  on  the  x-axis  is  equal  to  the  imit  segment  1  to  10  on  the 
i/-axis. 

Constructing  the  table  of  values  and  plotting  the  points  as  usual, 
we  have  the  graph  in  the     Y 
figure. 


X 

y 

0 

3.00 

0.5 

5.33 

1.0 

9.49 

Taking  the  logarithms  of 
both  sides  of  the  equation 
y  =  3-10°-5*,  we  obtain  the 
equation 

log  y  =  \ogS  +  0.5a;, 
or 

F  =  0.5x  +  log3, 

whose  graph  is  a  straight  line  referred  to  the  uniform  scales  on  the  axes. 
Hence  the  graph  oi  y  =  3*  10°*^*  plotted  on  the  semi-logarithmic  paper  is  the 
^  straight  line  whose  slope  is  0.5  and  whose  intercept  on  the  2/-axis  is  log  3. 
Note  that  the  uniform  scales  determine  the  slope. 

Theorem  2.     The  graph  of  the  exponential  function  y  =  fclO*"* 
b  plotted  on  semi-logarithmic  paper  is  a  straight  line  whose  slope 
is  m  and  whose  intercept  on  the  y~axis  is  k. 
The  proof  of  this  theorem  is  left  as  an  exercise. 


256 


ELEMENTARY  FUNCTIONS 


EXERCISES 

1.  Make  a  paper  slide  rule  by  copying  the  logarithmic  scale  given  in 
Fig.  144  on  two  paper  rules  of  an  adequate  degree  of  stiffness.  With  this 
instrument  calculate  the  following  and  describe  the  process. 

4  X  6,  I,  f,  6  X  9,  (5  X  7)/(8  x  9). 

2.  Write  the  equations  of  the  lines  a,  6,  c,  .  .  .  g,  in  Fig.  150. 

3.  Write  the  equations  of  the  lines  in  Fig.  152. 

4.  Plot  the  graphs  of  the  following  equations  on  logarithmic  paper. 


(a) 

2/  =  ar», 

(b) 

y  -  3x2, 

(c) 

y-2x\ 

(d) 

y  -  hx^ 

(e) 

y  =  3x, 

(f) 

4 

(g)  pv  =  4, 


(h)  pv 


vr 


A    .5    .6    .7    .8 

Fig.  152. 


6.  Plot  the  graphs  of  the  following 
equations  on  semi-logarithmic  paper. 

(a)  2/ -2(102-),  (b)  2/ =  3(10-^) 

(c)  y  =  K10-),         (d)  y  =  e^. 

6.  Measure  the  length  of  the  loga- 
rithmic scale  in  Fig.  144  and  calculate 
the  radius  of  a  circle  whose  circumfer- 
ence is  equal  to  this  length.  Fasten 
a  circular  disk  of  this  radius  to  a  larger 
disk  by  means  of  a  pin  through  their 
centers  so  that  they  may  rotate  freely.  On  the  circumference  of  the 
smaller  disk  and  on  the  circle  of  equal  radius  of  the  larger  copy  the 
logarithmic  scale  of  Fig.  144.  Make  the  calculations  of  Exercise  1  with 
this  instrument.  What  advantage  has  the  circular  slide  rule  over  the 
slide  rule,  of  the  first  exercise? 

7.  Find  the  equations  of  the  lines  in  Fig.  150  if  the  given  ranges  of  the 
scales  are  as  follows: 

(a)  X-axis,  10  to  100,  (b)    0. 1  to  1,  (c)  0.01  to  0. 1, 

y-axis,    1  to    10.  10.0  to  100.  0.1    to  1.0. 

8.  Find  the  equations  of  the  lines  in  Exercise  3  if  the  ranges  of  the  scales 
are  as  follows: 

(a)  10  to  20,  (b)  -5toO,  (c)      4  to  8, 

0. 01  to  0. 1.  10  to  100.  100  to  1000.       \ 


X-BXIS, 

j/-axis, 


90.  Empirical  Data  Problems.    If  the  points  representing  a! 
given  table  of  empirical  data  appear  to  lie  on  the  graph  of  I 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     257 


an  equation  of  the  form  y  =  klO""'^  the  constants  k  and  m  may- 
be determined  as  in  the 

Example.     Determine  the  law  representing  the  table,  and  find  the 
value  of  y  ii  X  =  2.5. 

Plotting  the  points  whose  coordinates  are    x\    1,        2,        3,        4 
the  pairs  of  values  in  the  table  we  get  a  curve    y  |2.37,  3.75,  5.97,  9.45 
which  appears  to  be  of  the  form 

2/  =  fclO^.  (1) 

To  test  this  assumption,  we  plot  the  table  on  semi-logarithmic  paper. 
As  the  points  obtained  lie  very  nearly  on  a  straight  line  we  conclude  that 
the  law  may  be  represented  by  an  equation  of  the 
form  (1)  with  a  fair  degree  of  accuracy  (Theorem  2, 
Section  89). 


y 

—9 

p 

1 

—7 
-6 

1 

/ 

/ 

—3 

—a 

—1 

/ 

f 

j 

/ 

0 

' 

■t 

Y 

r 

y 

10 
6 

1 

^__^..^'^ 

'*'* 

^ 

"jrl^ 

0 

' 

i 

>           ,< 

?           ^ 

I         X 

Fig.  163. 


Fig.  154. 


?o  determine  fc  and  m,  equate  the  logarithms  of  both  sides  of  (1),  which 


(2) 
(3) 


log  y  =  tnx  4-  log  fc 
or  Y  =  mx-\-K, 

where  F  =  log  y        and        K  =  log  k. 

Equation  (3)  is  linear  in  the  variables  x  and  Y,  and  the  constants  m 
and  K  can  be  determined  by  the  method  in  Section  27,  page  78,  from  a 
table  of  values  of  x  and  Y.  We  therefore  look  up  the  logarithms  of  the 
given  values  of  y,  which  are  the  values  of  Y.  From  this  table  we  obtain, 
by    the    method    referred     to,    the    values 


=  0.175. 
to   the 


Since  K  =  \ogk  '=    — - 


1,        2, 


3, 


m  =  0.200  and  iC      _  .. 

#0.175,  reference  to  the  table  shows  that  ^  '  "^'^'  '^^^'  ' ' '^'  '^'^ 
k  =  1.49.  Substituting  the  values  of  k  and  m  in  (1),  we  have  the  required 
law, 

!/  =  l.  49(10°- 20to)  (4) 

If  X  =  2.5,  2/  -  1.49(100-200  X  2.5)  =  4.73. 

If  the  points  representing  a  given  table  appear  to  he  on  the 
graph  of  an  equation  of  the  form  y  =  A;x"  (page  119),  a  value  of 
n  may  be  chosen  by  means  of  the  form  of  the  graph,  and  the 


258  ELEMENTARY  FUNCTIONS 

value  of  k  determined,  as  in  Section  44,  page  127.  The  method 
is  unsatisfactory  in  that  the  only  way  to  tell  which  of  two 
values  of  n  is  the  better,  for  example,  whether  n  =  2  or  n  =  | 
is  the  better,  is  to  try  them  both.  The  following  procedure  is 
preferable. 

If  the  points  representing  the  table  appear  to  lie  on  the  graph 
of  an  equation  of  the  form 

y  =  kx^'y  (5) 

plot  the  points  on  logarithmic  paper.  If  the  graph  is  now  ap- 
proximately straight  (Theorem  1,  Section  89),  we  conclude  that 
the  law  may  be  well  represented  by  (5).  As  the  logarithms  of 
the  two  sides  of  (5)  must  be  equal,  we  have 

logy  =  n  log  X  +  log  kj  (6) 

or  Y  =  nX  +  K,  (7) 

where  Y  =  \ogy,  X  =  log  x,  and  K  =  log  k. 

With  the  table  of  logarithms  build  the  table  of  values  of  X 
and  Y  corresponding  to  the  given  table  of  values  of  x  and  y. 
From  it  the  values  of  n  and  K  may  be  found  by  the  method  in 
Section  27,  page  78.  The  value  of  k  is  then  found  from  that 
of  K. 

It  should  be  noticed  that  if  we  suspect  that  the  law  has  the 
form  (1)  or  (5)  it  is  not  essential  that  the  table  should  ba 
plotted  on  semi-logarithmic  or  logarithmic  paper,  respectively. 
But  this  plotting  strengthens  our  feeling  of  having  a  suitable 
law,  and  approximate  values  of  the  constants  may  be  obtained 
easily  directly  from  the  gra  ph  (see  Example  3,  Section  89,  and 
Exercises  2  and  3  of  the  preceding  set).  Of  course,  if  the  new 
graph  is  not  approximately  straight  the  choice  of  one  of  these 
two  laws  does  not  give  a  very  good  representation  of  the  given , 
data. 

If  the  points  representing  the  table,  plotted  on  ordinary  cross- 
section  paper,  appear  to  he  on  the  graph  of 

y  =  clog  nx,  (8) 

we  interchange  x  and  i/,  and  proceed  as  in  the  example  above. 
The  graph  of  (5)  may  be  distinguished  from  that  of  (1)  or 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS    259 

(8)  by  the  fact  that  it  passes  through  the  origin,  if  n  is  positive, 
and  that  it  does  not  cut  either  axis,  if  n  is  negative. 


EXERCISES 

1.  Find  the  law  of  growth  of  the  population  of  the  United  States  from 
the  following  data: 

Year  1  1880, 1890, 1900^ 1910, 

Population  in  millions  |     50,  62,  76,  92, 

Let  X  =  0  in  1880.     Assuming  the  law  of  growth  does  not  change  find  the 
population  in  1920. 

2.  Water  flows  out  of  a  sharp-edged  circular  opening  in  the  vertical 
side  of  a  tank.  The  table  gives  pairs  of  values  of  the  head  h,  the  height 
in  feet  of  the  surface  of  the  water  above  the  center  of  the  opening,  and  the 
discharge  Q,  the  number  of  cubic  feet  of  water  flowing  through  the  opening 
in  a  second.     Determine  Q  as  a  function  of  h. 

h  I    0.390,      0.709,       1.013,       1.540, 
Q  I    0.0376,    0.0505,    0.0602,    0.0740, 

3.  A  rectangular  weir  is  a  rectangular  notch  in  the  wall  of  a  reservoir 
or  channel.  The  table  gives  pairs  of  values  of  the  head  h,  the  height  in 
feet  of  the  surface  of  the  water  above  the  bottom  of  the  notch,  and  the  dis- 
charge Q  in  cubic  feet  per  second.    Find  the  law.     Find  Q  if  ^  =  0.265. 

h  I    0.053,      0.103,      0.160,      0.217, 
q\    0.0422,     0.1095,     0.2065,     0.3200, 

4.  A  triangular  weir  is  a  triangular  notch  in  the  side  of  a  tank  or  channel. 
'The  table  gives  pairs  of  values  of  the  head,  h,  and  the  discharge,  Q.     The 

angle  of  the  notch  in  this  case  was  90°.     Find  the  law. 

h\     0.095,         0.169,       0.249,       0.330,       0.392, 
q\     0.00615,     0.0302,     0.0780,     0.1590,    0.2445, 

5.  In  a  chemical  experiment  it  was  found  that  the  concentration  x  of 
a  solution  was  connected  with  the  amount  of  precipitation  y  of  a,  metal 
as  in  the  table.     Determine  the  law  and  find  y  ii  x  =  5. 

X  \     1,    2,    4,      8, 


• 


2/  I     1,     3,     9,     27, 


Find  the  velocity,  v,  of  a  certain  chemical  reaction  as  a  function  of 
the  temperature,  T,  from  the  data  given  in  the  table.     Find  v  ii  T  =  50°. 

T  I      0°,         10,        20,        30,         40, 


V  \     1.00,     2.08,     4.32,     8.38,     16.19, 


260  ELEMENTARY  FUNCTIONS 

7.  Find  the  law  connecting  the  maximum  speed,  v,  of  an  electric  vehicle 
and  the  total  weight,  W,  in  thousand  pounds  from  the  following  data. 
What  weight  will  reduce  the  maximum  to  6  miles  per  hour? 

W  \     2.0,     3.0,     4.2,     6.5,     8.5, 
V  I      20,       14,       12,       10,      .8, 

8.  Find  the  law  connecting  the  cost,  C,  in  cents  per  miles  for  tires, 
repairs,  battery,  electricity,  of  electric  vehicles  of  weight,  W,  in  thousand 
pounds.     What  will  be  the  cost  per  mile  of  a  vehicle  weighing  10,000? 

W  \       1,        2,         4,        7, 
C\     7.9,     9.1,     11.5,     16.0, 

9.  From  the  table  find  the  law  connecting  the  resistance,  R  (in  pounds 
per  ton),  which  a  passenger  train  encounters  at  a  speed,  v  (miles  per  hour). 
Find  i2  if  y  =  60. 

V  I     5,         10,       15,      20,      25,      35, 
r\    5.9,    5.5,     5.4,     5.5,    5.6,     6.2, 

10.  Airships  are  provided  with  air  bags  within  their  hulls  for  regulat- 
ing the  height  of  ascent.  The  total  cubic  contents  of  these  air  bags  when 
filled  is  V  =  mV  where  V  is  the  cubic  contents  of  the  airship.  Values 
of  m  for  various  heights,  H,  are  given  in  the  table.  Find  the  law  and  the 
value  of  m  when  H  =  8. 

H  in  thousand  ft.  1     1,  2, 4, 10^ 

m=V'/V  I  0.04,     0.075,     0.141,     0.318, 

11.  Find  the  law  connecting  the  load,  L  (in  pounds  per  square  foot  of 
wing)  of  an  aeroplane  with  the  area,  A,  of  wings  (in  square  feet).  Find 
the  safe  load  ii  A  =  200. 

A  I  150,     300,     400,     700, 
L  I  9.4,    6.8,    6.2,    5.1, 

MISCELLANEOUS  EXERCISES 

1.  The  temperature  of  a  body  cooling  according  to  Newton's  law, 
6  =  6(fi~^\  fell  from  125°  to  94**  in  8  minutes.  Find  the  equation  con- 
necting the  temperature  and  the  time  of  cooling. 

2.  Construct  a  scale  for  the  function  x^  following  the  method  used  in 
constructing  the  logarithmic  scale.  Find  by  means  of  the  scale  (2.3)' 
and  VTT. 

3.  Construct  a  table  of  values  for  the  function  y  =  2x''  +  3  and  plot 
the  graph.  Let  u  =  x^  and  hence  y  =  2u  -\-  Z.  Construct  on  the  hori- 
zontal axis  a  uniform  scale  of  u  and  a  corresponding  non-uniform  scale  of 
X.  Plot  the  table  of  values  of  x  and  y  using  the  non-uniform  scale  of  x 
and  the  uniform  scale  of  y.    What  is  the  character  of  the  graph? 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     261 


4.  Determine  scales  on  the  axes  so  that  the  graph  oi  y  =  S{1QP'°^  )  is  a 
straight  line. 

5.  Determine  the  constants  my  =  /clO'"*^  by  means  of  the  pairs  of  values 
(2,  0.5)  and  (3,  0.8).  Choose  scales  on  the  axes  so  that  the  graph  of  the 
equation  is  a  straight  line.     From  the  graph  read  the  value  of  t/  if  x  =2.5. 

6.  Find  the  radius  of  the  moon  in  miles,  given  that  the  diameter  of  the 
moon  subtends  an  angle  of  32'  as  seen  from  the  earth,  the  distance  of 
the  moon  from  the  earth  being  240,000.  Find  the  ratio  of  the  mass  of  the 
moon  to  that  of  the  earth  if  the  density  of  the  moon  is  0.6  that  of  the 
earth. 

7.  If  the  distance  of  the  sun  from  the  earth  is  92,800,000  miles  and  the 
diameter  of  the  sun  subtends  an  angle  of  32.4'  as  seen  from  the  earth, 
find  the  diameter  of  the  sun,  and  the  ratio  of  the  volume  of  the  sun  to  that 
of  the  earth. 

8.  If  the  length,  I,  of  a  range  finder  is  20  feet  and  the  distances,  d,  are 
calculated  by  the  formula  d  =  I  tan  6,  find  the  values  of  6  corresponding 
to  the  extreme  values  400  ft.  and  20,000  ft.  on  the  dial  of  the  instrument. 
(Use  the  "  log  rad  "  table  on  page  28  of  the  Tables.) 

9.  The  following  table  gives  the  collegiate  running  records,  the  distance 
d  being  in  yards  and  the  lime  t  in  seconds.  Find  the  law  and  the  value  of 
t\id  =  600.     Find  f  if  d  =  3520. 

d  I     100.      440,      880,     1760. 


t  I     9.75,     47.4,     113,     254.4, 


10.  An  observer  on  a  destroyer  moving  at  the  rate  of  35  miles  an  hour 
notes  that  the  line  of  sight  to  a  ship  makes  an  angle  of  41°.56  with  the  for- 
ward path  of  the  destroyer  and  that  one  minute  later  the  angle  is  74°.83. 
He  estimates  that  the  ship  is  moving  on  a  parallel  path  at  the  rate  of 
14  miles  an  hour.    Find  the  distance  between  the  paths. 

11.  The  economic  law  of  diminishing  utility  is  stated  as  follows: 
The  total  utility  of  a  thing  to  any  one  (that  is,  the  total  pleasure  or  other 

benefit  the  thing  yields)  increases  with  any  increase  in  one's  stock  of  it 
but  not  so  fast  as  the  stock  increases.  If  one's  stock  increases  at  a  uniform 
rate,  the  benefit  derived  from  it  increases  at  a  diminishing  rate. 

Another  way  of  stating  the  law  is:  The  increase  in  the  utility  of  a  thing, 
or  marginal  utility,  diminishes  with  every  increase  in  the  amount  of  it 
any  one  already  has. 

Plot  the  graphs  of  these  two  statements  of  the  law. 

12.  Find  approximate  values  of  the  real  roots  of  the  following  equa- 
tions, which  can  not  be  solved  by  the  ordinary  methods. 

Hint.  Plot  the  graph  of  the  function  on  the  left  and  from  it  locate 
an  intersection  with  the  a;-axis  as  accurately  as  possible.  Enlarge  the 
table  of  values  until  the  coordinates  of  two  points  are  obtained  such  that 
the  intersection  lies  between  them  and  such  that  the  part  of  the  graph 


262 


ELEMENTARY  FUNCTIONS 


2-. 


Ix 


m 


between  them  may  be  assumed  straight.     Then  determine  an  approximate 
value  of  the  root  by  interpolation. 

(a)  fix)  =  2~*  -  ix  =  0.  Solution.  The  graph  of  f{x)  shows  that 
there  is  a  root  between  1  and  2,  near  1.2.  Enlarging  the  table  of  values, 
it  is  seen  that  the  root  is  between  1.2  and  1.3,  since  the  corresponding 
values  of  f{x)  have  opposite  signs. 

In  order  to  find  a  value  of  x  by  interpolation, 
construct  the  graph  between  x  =  1.2  and  x  =  1.3 
on  the  assumption  that  it  is  a  straight  line  BD. 
To  find  AE  we  have,  by  similar  triangles, 
AE^AB 
6  BF~  DF' 

'---     Substituting  BF  =  AC  =  0.1, 
0  035  AB  =  0.035, 

-0.027  and  DF  =  0.027  +  0.035  =  0.062, 


-1 
0 
1 
2 


1.2 
1.3 


2 
1 

_i 

0.435 
0.406 


0.400 
0.433 


2i 

1 


""- 

^ 

_,s 

^v 

^^^           2 

\S 

x\ 

eSv 

^^^ 

^ 

1^^                                              L 

5^^          iki- 

^5^"^    - ^^ 

^-'•'            S^                                ^ 

~          ^  '  0                            V        2            or 

: :_  :_± 

we  get 
whence 


AE 
0.1 


0.035 


0.062 
AE  =  0.056. 


=  0.56, 


Then  the  abscissa  of  ^  is  x  =  1.2 
+  0.0c6  =  1.256,  which  is  an  approxi- 
mate value  of  the  root  of  the  given 
equation. 


.035 

"\' 

-— 1 

1 

1 
1 

A 

^*"**^,^ 

C       X 

1 

£^^N 

■->^-.027 

Fig.  156. 

(d)  2-»  -  x/2 

-1=0. 

=  0. 

(g)  2  sin  X  -  3x/2  =  0. 

). 

(j)  sin  X  -  x"^  = 

=  0. 

Fig.  1-55. 

(b)  2-*  -  x      =  0.         (c)  3-*/2  -x  =  0. 
(e)  4-*-a;/5  =  0.         (f)    sin  x  -  3a;/4 
(h)  cos  X  -  X  =  0.        (i)    tan  a;-  2x  =  0 
(k)  2~*  -  sin  X  =  0  (smallest  positive  root). 
(1)    cos  X  -  2~*  =  0  (smallest  positive  root), 
(m)  e-*  -  a;  =  0.  (n)  e"*  -  x/3  =  0. 

13.  Some  properties  of  many  functions  can  be  expressed  entirely  in  terms 
of  the  notation  f(x) .  If  analogous  properties  of  two  functions  can  be  so  ex- 
pressed an  abstract  point  of  view  is  obtained  which  gives  a  deeper  insight 
into  the  differences  between  the  functions.  Establish  the  following 
relations:  » 


(o)  e-*/2  -  x/3  =  0. 


EXPONENTIAL  AND  LOGARITHMIC  FUNCTIONS     263 


(a)  If  J{x)  =  x^  or  cos  x, 
If  J{x)  =  a:'  or  sin  x, 

(b)  If/(x)=rc«, 
If  fix)  =  log  X, 

(c)  If/(x)=x«, 


fi-a)-m. 
/(-a)  =  -  /(a). 

fiab)=f{a)xm. 

f(p)_f(A 


m 


ft). 


(d)  If /(a:)  =  Vx,  kj{a)  ^f{k%), 

Jifix)=logx,  fc/(a)=/(a*). 

14.  The  speed  of  an  aeroplane  is  80  miles  an  hour  and  the  wind  is  blow- 
ing from  the  north  with  a  velocity  of  20  miles  per  hour.  The  pilot  desires 
to  move  S.  E.  Find  the  direction  in  which  he  should  head  the  machine 
and  how  fast  he  will  move. 

15.  Solve  the  preceding  exercise  supposing  that  the  pilot  desires  to  sail 
N.  W.  If  the  aeroplane  can  stay  in  the  air  4  hours,  find  the  greatest  dis- 
tance the  pilot  can  sail  S.  E.  and  be  able  to  return  to  the  starting  point. 

16.  The  speed  of  an  aeroplane  is  100  miles  per  hour  and  the  wind  is 
blowing  from  the  west.  The  plane  can  stay  in  the  air  5  hours.  How  far 
can  the  pilot  sail  in  the  direction  10°  S.  of  W.  and  return  to  the  starting 
point? 


CHAPTER  VI 
DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS 

91.  Introduction.  If  ?/  is  a  function  of  x,  the  average  rate  of 
change  of  y  with  respect  to  x  in  any  interval  Ax  is  Ay  /Ax  (page 
35).  If  the  average  rate  of  change  is  constant,  its  value  is  the 
rate  of  change  of  y  with  respect  to  x  (Definition,  page  48), 
and  the  graph  of  2/  is  a  straight  line  (Theorem,  page  50)  whose 
slope  is  the  rate  of  change.  Numerous  appHcations  of  uniform 
rate  of  change  were  given  in  Chapter  IT. 

If  the  average  rate  of  change,  Ay /Ax,  is  not  constant,  then 
the  rate  of  change  of  y  with  respect  to  x  is  defined  to  be  the 
limit  of  Ay /Ax  as  Ax  approaches  zero  (page  94).  It  is  repre- 
sented graphically  by  the  slope  m  of  a  fine  tangent  to  the  graph 
of  the  function  y. 

In  this  chapter  we  shall  consider  this  limit  more  formally 
than  heretofore,  and  derive  rules  for  finding  it  expeditiously 
if  y  is  an  algebraic  function  of  x.  The  appHcations  are  based 
either  on  the  geometric  interpretation  of  the  limit  of  Ay /Ax  as 
the  slope  of  the  tangent  line  or  on  the  physical  interpretation 
of  the  limit  as  the  rate  of  change  of  y  with  respect  to  x. 

The  ideas  to  be  considered  in  this  chapter,  and  the  one 
following,  are  among  the  most  fundamental  and  far-reaching 
concepts  in  mathematics.  They  were  developed  by  the  famous 
Englishman  Sir  Isaac  Newton  (1642-1727)  and  the  noted 
German  Gottfried  Wilhelm  von  Liebnitz  (1646-1716),  and  a 
bitter  controversy  lasting  for  many  years  was  waged  over  the 
question  as  to  which  one  of  these  men  should  be  accorded  the 
honor  of  the  discovery.  Leibnitz  was  the  first  to  publish  some 
of  his  results,  in  1684,  but  Newton  had  written  a  paper  on  the 
subject  and  submitted  it  to  some  of  his  friends  in  1669.    The 

264 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    265 


I 


followers  of  each  claimed  that  the  other  had  been  guilty  of 
claiming  ideas  not  his  own,  but  most  historians  of  mathe- 
matics are  agreed  that  the  work  of  Newton  and  Leibnitz  was 
independent.  On  the  basis  of  the  work  of  these  men,  there 
followed  a  period  of  rapid  and  extensive  mathematical  develop- 
ment. 

»!  Let  us  first  consider  the  underlying  concept  of  the  limit  of 
a  variable. 
92.  Limits.  By  definition  (footnote,  page  93)  the  limit  of  a 
variable  x  is  sl  constant  such  that  the  numerical  value  of  the 
difference  between  x  and  a  becomes  and  remains  as  small  as 
we  please.  This  explains  what  is  meant  by  saying  that ''  x  ap- 
proaches a  as  a  limit,"  or  in  more  compact  form,  '^  x  approaches 
a."    It  is  immaterial  whether  or  not  x  becomes  equal  to  a. 

A  function  of  a:  is  a  second  variable  y  (Definition,  page  5). 
The  functions  we  have  studied  are  such  that  if  x  approaches  a 
limit,  so  also  does  y,  provided  that  y  does  not  become  infinite 
as  X  approaches  a.    The  notation 
Hr  hmy  =  b  (!) 

is  used  to  indicate  that  "  the  limit  of  y,  bs  x  approaches  a,  is 
6."  This  means  that  the  numerical  value  of  the  difference  be- 
tween y  and  b  can  be  made  as  small 
as  we  please  by  taking  x  sufficiently 
near  to  a. 

Graphically,  the  difference  between 
the  ordinates  y  and  h  can  be  made  as 
small  as  we  please  by  making  the 
;  difference  between  the  abscissas  x 
and  a  sufficiently  small.  In  other 
words,  Ay  =  b  -  y  approaches  zero 
ii  Ax  =  a  —  X  approaches  zero. 

For  example,  the  average  rate  of 
change  of  the  function  ax^  in  an  interval  Ax  beginning  at  a 
definite  point  x  is  (see  (5),  page  95) 
Ay 


FiQ.  157. 


Ax 


2ax  +  a  Ax, 


266  ELEMENTARY  FUNCTIONS 

The  rate  of  change  of  y  with  respect  to  x  for  a  given  value  of 
X  is  therefore 

m  =  Um   -T^  =  Hm  (2ax  +  aAx). 

Ax=0   i^X        Aa;=0 

We  are  thus  led  to  find  the  limit  of  2ax  +  a  Ao;,  a  function  of 
Ax,  as  Ax  approaches  zero.  In  computing  this  limit,  x  has 
a  given  value  and  is  regarded  as  constant. 

In  order  to  prove  the  assumption  made  earher  that  the 

limit  is  2ax,  it  must  be  shown  that  a  Ax,  the  difference  between 

the  variable  2ax  +  a  Ax  and  the  constant  2ax,  can  be  made  as 

small  as  we  please  by  making  Ax  sufficiently  small.    This 

follows  readily.     For  if  we  wish  to  make  a  Ax  as  small  as  0.001, 

it  is  sufficient  to  take  Ax  =  0.001  /a,  which  is  possible  since 

Ax  approaches  zero  and  can  therefore  be  made  as  small  as  we 

please. 

Ay 
The  limits  encountered  in  computing  lim    -r- ,  where  y  is  a, 

Ax=0  ^X 

polynomial  or  a  rational  function  of  x,  can  be  computed  by 
means  of  the  following  theorems,  which  we  assume  without 
proof. 

Theorem  1.  The  limit  of  the  sum  of  several  variables  is  the 
sum  of  their  limits. 

Theorem  2.  The  limit  of  the  product  of  several  variables  is  the 
product  of  their  limits. 

Theorem  3.  The  limit  of  the  quotient  of  two  variables  is  the 
quotient  of  their  limits,  provided  the  limit  of  the  divisor  is  not 
zero. 

The  limit  of  the  difference  of  two  variables  may  be  found  by 
Theorem  1,  since  u  -  v  =  u  +  (-  v). 

In  applying  these  theorems,  it  is  frequently  convenient  to 
regard  ?/  =  c,  a  constant,  as  a  function  of  x,  whose  limit,  as  x 
approaches  a,  is  c. 

Example.    If  y  >=  x^,  then 

Ax 

Give  the  details  of  the  computation  of  the  limit  of  Ay/ Ax  as  Ax  approach* 
sere. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    267 


Denoting  the  limit  by  w,  we  have 
m  =    lim     ^  =    lim    (3x2  +  3x  Ax  +  ^x'^) 


=    hm    3x2  ^   lim  3a;  ^  ^  ]^j^   ^  ^   (Theorem  1) 
Ax=0  Ax=0  Ax=0 

=  3a;2  +     lim    Zx    hm     Ax  +     hm     Ax    hm  Ax 
Ax=0  Ax=0  Ax=0  Ax=0 

(Theorem  2) 


=  3x2 


(Since    lim  Ax  =  0) 
Ax=0 


93.  Derivative  of  a  Function.  Definition.  If  2/  is  a  func- 
tion of  X,  and  if  At/  is  the  change  in  y  corresponding  to  a 
change  of  Aa;  in  x,  then  the  limit  of  Ay  /Ax,  as  Ax  approaches 
zero,  is  called  the  derivative  of  y  with  respect  to  x.  Denoting  it 
by  Dxy  (read  "  the  derivative  of  y  with  respect  to  a:  '0;  the 
definition  may  be  expressed  by  the  equation 

le  results  obtained  in  Section  33,  page  94,  may  now  be 

ited  as  follows: 

\The  derivative  of  y  with  respect  to  x,  Dxy,  measures  the  rate  of 

mge  of  y  with  respect  to  x. 

The  derivative  of  y  with  re- 
spect to  X  is  represented  graphi- 
cally hy  the  slope  of  a  line  tangent 
to  the  graph  of  y.     That  is 


Dx2/ =  m  =  tan  0.        (2) 

jj     The  process   of   finding   the 

If  derivative  of  a  function  is  called 
differentiation,  and  the  succes- 
sive steps  in  the  process  have  been  given  in  the  section  cited 
above. 


Example.     Differentiate  y  = 


Substituting  x  +  Ax  f or  x  and  y  +  Ay  for  y, 

1 


y+  Ay 


(x  +  Ax)2 


268  ELEMENTARY  FUNCTIONS 

Subtracting  the  value  of  y, 

_         1         _  L  _  a^^  -  (a^  +  Ax)^      -2x  Ax- Axy 
^  ~  (a;  +  ^xY     x^  ~    x\x  +  Aa;)2    "    x^a:  +  Ax)^ 

Dividing  by  Ax, 

Ay  _  —  2x  -  Ax 

Ax  ""  x2(xTax)2* 

Passing  to  the  limit  as  Ax  approaches  zero, 

lim  (-  2x  -  Ax) 

D      =   hm     ^  =   lim     "  ^^  "  ^^  ^  Ax==o 

*^       Ax=0  Ax      Ax=0  x^ix  +  Ax)2  "    lim  x2(x  +  Ax)2 

Ax=0 

by  Theorem  3  of  the  preceding  section.     Applying  Theorem  1  in  tl 
numerator  and  Theorems  1  and  2  in  the  denominator,  we  get 

n     _      2±-      2 
^^y  -  "  x2x2  ~  "  x3* 


EXERCISES 

1.  Evaluate  the  following  limits,  indicating  in  detail  the  use  of  tl 

theorems  in  Section  92. 

(a)  lim       (2-3x  +  x2).  (b)  lim     a;' -  3x 

x=0  x=2        x+l 

(c)  lim       (2x  +  Ax  -  4) .  (d)      Hm     (3x2  +  3a;  ax  +  Ax^  -  6x  -  3  Ax)il 

AX  =  0  AX:S:0 

(e)     lim    x2  +  X  Ax  +  1  (f)     lim    2x+ Ax-x2-xAx 

Ax=0      x{x  +  Ax)     *  Ax=0        x2(x  +  Ax)2 

2.  Differentiate  the  functions: 

(a)  y  =  x3  -  3x.  (b)  y  =  1/x.  (c)  y  =  (x  +  l)/x. 

(d)  y  =  c.  (e)  2/  =  x.  (f)  mx  +  6. 

(g)  y  =  y/x.    Hint.    Rationalize  the  numerator  of  Ay /Ax  before  passi 
to  the  limit  as  Ax  approaches  zero. 

3.  If /(x)  =  (x2  +  3x)/(x2- 1),  find  lim   /(x).     Note    that    the    value 

obtained  is  f(a),  provided  o  5^  ±  1.     What  happens  if  o  =  :*=  1? 

4.  If  fix)  =  x2,  prove  the  relations 

lim  fix)  -  fid)        and        lim     Ay  -=  0 

x^a  Az^O 

and  interpret  them  graphicall}- . 

6.   Prove  that  the  relations  in  Exercise  4  are  true  if  fix)  is  any  qi 
ratic  function. 


5.  . 

I 


DlFFEREMTiATlON   OF    Al.GEBRAlC   FUNCTIONS     269 

6.  If  f{x)  is  any  function,  and  Ax  =  a  -  x,  show  that  the  first  relation 
in  Exercise  4  is  true  if  and  only  if  the  second  is  true.  What  is  the  graphi- 
cal significance  of  each  relation? 

Note.  A  function  is  said  to  be  continuous  at  x  =  a  ii  the  first  relation 
in  Exercise  4  is  true  for  the  given  function.  It  can  be  proved  that  the 
algebraic  and  transcendental  functions  studied  in  this  course  are  con- 
tinuous for  all  values  of  x  for  which  they  do  not  become  infinite.  If  a 
function  becomes  infinite  as  x  approaches  a,  then  f{a)  has  no  meaning. 
Hence  x  =  a  is  a  point  of  discontinuity.  There  are  other  types  of  points 
of  discontinuity. 

7.  Prove  that  (a)  any  polynomial,  (b)  any  rational  function,  is  con- 
tinuous for  all  values  of  a,  except,  in  (b),  for  the  values  for  which  the  func- 
tion becomes  infinite. 

/\u 

8.  If  w  is  a  function  of  x,  what  is  the  value  of    lim  Aw?  of   lim    — — ?  of 

AxaO  Ax=o  Ax 

,.       Au^       ,.        .     Aw„ 
lim    — —  =   lim    Au  -T— ? 
Ax=0  Ax       Ax=0         Ax 


b  94.  Fundamental  Formulas  for  Differentiation.  The  rules  in 
this  section  are  useful  in  differentiating  a  function  without  the 
labor  involved  in  computing  the  limit  considered  in  Section  93. 
I    Theorem  1.     The  derivative  of  a  constant  is  zero;  that  is, 


D:,C   =  0. 


(1) 


For  the  graph  of  i/  =  c  is  a  straight  Hne  parallel  to  the  x-axis, 
whose  slope,  m  =  Dxy  =  DxC,  is  zero. 

Theorem   2.     The  derivative  of  the  in- 
dependent variable  is  unity,  that  is, 


D^x  =  1. 


(2) 


For  the  graph  oi  y  =  x  is  the  straight 
line  bisecting  the  first  and  third  quad- 
rants, whose  slope,  m  =  Dxy  =  DxX,  is 
unity. 

Theorem  3.     The  derivative  of  a  constant 
times  a  function  is  equal  to  the  constant  times  the  derivative  of 
the  function.    Symbolically,  if  u  is  any  function  of  x, 


Fig.  159. 


Let 


DxCU  =  cDxU. 

y  =  cu. 


(3) 


270  ELEMENTARY  FUNCTIONS 

Then  y  +  Ay  =  c{u  +  Au), 

and  hence,  subtracting,  Ay  ==  c  Au. 

Dividing  by  Ax,  S  =  ^S* 

XT  T^  T       Aw       T  Au  ,.       Au       T^ 

Hence   Dxy  =  hm    -r^  =  hm    c  -r-  =  c   um    -r—  =cDxU. 

Ax=o  Ax      Ax=o      Ax         Ax=o  Ax 

What  is  the  graphical  interpretation  of  this  theorem,  in  the 
Ught  of  the  Theorem  on  page  89? 

Theorem  4.  The  derivative  of  the  sum  of  two  functions  is  equal 
to  the  sum  of  their  derivatives.  Symholically,  if  u  and  v  are  any 
two  functions  of  x 

Dx{u  +  y)  =  D^u  +  DxV.  (4) 

Let  y  =  u  +  V. 

Then  y  -\-  Ay  =  u  +  Au  +  v  -{■  Av, 

and  hence  Ay  =  Au  +  Av. 

•r^-  •  T      1      A  Av      Au      Av 

Dividing  by  Ax  Sc^Ai+M' 

Therefore     D.y  =  ^n^   g  =  jim     (^  +  ^) 

,.        Aw  ,    ,.       At;      y.       ,  y^ 
=  hm    -7 — h  hm    -r-  =  D^u  +  D^v, 

Ax=iO    ^^        Ax=0    ^a; 

Corollary.  The  derivative  of  the  sum  of  several  functions  is 
the  sum  of  their  derivatives. 

Theorem  5.  The  derivative  of  the  nth  power  of  a  function 
of  X  is  n  times  the  (n  -  \)st  power  of  the  function  times  the  deriva- 
tive of  the  function  with  respect  to  x.    Symholically,  if  u  is  any 

function  of  x 

Z)xi/"  =  nW"-^  DxU.  (5) 

Let  y  =  w". 

Then  y  +  Ay  =  {u  +  Au)"" 

.     =  w"  +  nw"-i  Au  +  ^^^  "  ^^  w"-2Aw2  +  •  •  •  +  Au", 
by  the  binomial  theorem.    Subtracting, 

Ay  =  nu^-^'Au  +  ^  ^V  ^^  w'^-^Aw^  +  •  •  •   +  Au\ 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    271 
Hence, 

A?/  ,  Au     n  (n  —  1)       „  Au  .  Au  .       , 

Ax  Aa;2  Ax  Ax 

and  therefore,  passing  to  the  Limit  as  Ax  approaches  zero, 

Dxy  =  nu''~WxU, 

Corollary.  The  derivative  of  the  nth  power  of  x  is  n  times  the 
in  -  l)st  power  of  x;  that  is, 

DxX''  =  nx«-i.  (5a) 

For,  by  (5),  DxX"  =  nx^'-^DxX  =  nx^-^  since  DxX  =  1,  by  (2). 

Example  1.   Differentiate  y  =  3^  +  Sx^  +  5. 

We  have                              I>x2/  =  -DxX^  +  D^Sx^  +  D^5  by  (4) 

=  3a:2  +  3D.X2  +  0  by  (5a),  (3),  (1) 

=  3x2  +  3-2x  by  (5a) 
=  3x2  ^  6^.^ 

Example  2.     Differentiate  y  =  (x^  +  4x  -  5)'. 

If  we  think  of  x^  +  4x  -  5  as  a  function  u,  the  given  equation  has  the  form 
y  =  y?.     Applying  (5). 

D^y  =  DM  +  4x  -  5)5 

=  3(x2  +  4x  -  5)2Dx(x2  +  4x  -  5) 
=  3(x2  +  4x  -  5)2[Dxx2  +  DAx  +  D^{-  5)] 
^^  =  3(x2  +  4x  -  5)2(2x  +  4). 

™  95.  Derivative  of  a  Poljmomial.    Any  polynomial  has  the 
form  (page  133) 

y  =  oox"  +  aix"-i  +  •   •   •   +  an-ix  +  a„. 
Then     D^l/  =  I>x(aoa:"  +  aix""^  +  •    •    •   +  an-ix  +  an) 

=  D:c(aoX-)  +  Z)x(aia;"-i)  +  •    •    •  +Dx(an-ia:)+Dxan 
by  the  corollary  to  Theorem  4,  Section  94, 
=  OoDxX''  +  aiOxX""-^  +  •    •    •    +  an-iDxX  +  0 

»by  Theorems  3  and  1,  Section  94. 
=  aonx""-^  +  ai{n  -  l)a;"-2  _|_  .    .    .   4.  an-i 
by  (5a),  Section  94. 
Hence, 

The  successive  terms  of  the  derivative  of  a  polynomial  may  he 
found  from  the  corresponding  terms  of  the  polynomial  by  multiply- 
ing the  coefficient  by  the  exponent  and  decreasing  the  exponent  by 


272  ELEMENTARY  FUNCTIONS 

one.    In  applying  this  rule,  the  constant  term,  whose  derivative 
is  zero,  may  be  regarded  as  the  coefficient  of  a^  =  1. 

For  example,  if  y  =  3x^  -  4x^  +  3a;  -  7, 

then  D^y  =  12x3  _  i2a;2  +  3. 

96.  Corresponding  Properties  of  a  Function,  its  Graph,  and 
its  Derivative.  The  derivative  is  useful  in  expressing  some  of 
the  properties  given  on  page  42,  and  some  other  properties  also. 
Rate  of  Change.  The  steepness  of  the  graph  at  any  point  is 
measured  by  the  slope  of  the  tangent  line,  and  the  rate  of 
change  of  the  function  is  measured  by  the  value  of  the  derivative. 
Changes  of  the  Function.  The  graph  rises  (or  falls)  to  the 
right  if  and  only  if  the  slope  of  the  tangent  line  is  positive  (or 
negative).     Hence 

A  function  increases  (or  decreases)  as  x  increases  if  and  only 
if  its  derivative  is  positive  (or  negative). 

Maxima  and  Minima.    A  line  tangent  to  the  graph  is  hori-_ 
zontal  if  and  only  if  its  slope  is  zero.     Hence, 

To  find  the  abscissas  of  the  points  at  which  the  tangent  line 
horizontal,  set  the  derivative  equal  to  zero,  and  solve  the  resulth 
equation,  Dxy  =  0,  for  x. 

The  roots  of  this  equation  may  be  values  of  x  for  which  the 
function  has  a  maximum  or  minimum  value. 

At  a  maximum  point,  the 
function  ceases  to  increase 
and  begins  to  decrease,  and 
hence  (see  Changes  of  the 
Function  above),  as  x  in- 
creases, the  derivative  must 
change  sign  from  positive  to 
negative. 

Similarly,    at    a    minimum 
point,  as  x  increases,  the  de- 
rivative must  change  sign,  from  negative  to  positive. 

If,  as  X  increases  in  the  vicinity  of  a  horizontal  tangent,  the 
derivative  does  not  change  sign,  the  point  of  contact  is  neither 
a  maximum  nor  a  minimum  point. 


y 

^       1 

/B\                        J 

/ 

^\     J^ 

J 

"       \J        « 

D 

Fig.  160. 

DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    273 

For  example,  the  graph  oi  y  =  x^  has  a  horizontal  tangent 
at  the  origin  (page  112),  since  Dxy  =  Sx^  =  0  ii  x  =  0.  But  as 
X  increases  through  x  =  0,  the  derivative  Sx^  does  not  change 
sign,  and  the  origin  is  therefore  not  a  maximum  or  minimimi 
point. 

Concavity.  The  curve  in  the  figure  is  concave  downward 
from  A  to  C.  If  the  tangent  line  at  A  rolls  along  the  curve 
until  the  moving  point  of  contact  reaches  C,  the  slope  m  de- 
creases, and  hence  Dxtn  is  negative  (see  Changes  of  the  Function 
above).  Similarly,  from  C  to  E  the  curve  is  concave  upward. 
The  slope  of  a  tangent  hne  roUing  from  C  to  E  increases,  and 
Dxm  is  positive.     Hence 

//  the  graph  is  concave  upward  {or  downward)  ^  Dxm  is  positive 
{or  negative).  It  is  also  true  that  the  curve  is  concave  upward 
(or  downward)  only  if  Dxm  is  positive  (or  negative). 

It  follows  that  a  point  at  which  the  tangent  line  is  horizontal 
will  he  a  minimum  point  if  Dxm  is  positive,  or  a  maximum  point 
if  Dxm  is  negative. 

Since  m  =  Dxy  is  the  rate  of  change  of  y,  and  since  m  in- 
creases or  decreases  according  as  Dxm  is  positive  or  negative, 
it  follows  that 

The  rate  of  change  of  a  function  increases  or  decreases,  as  x 
increases,  according  as  the  graph  of  the  function  is  concave  up- 
ward or  downward. 

Points  of  Inflection.  At  a  point  of  inflection  (Definition, 
page  139)  m  ceases  to  increase  and  begins  to  decrease,  or  vice 
versa.  Hence  m  has  a  maximum  or  minimum  value,  and  there- 
fore Dxm  =  0  (see  Maxima  and  Minima  above).  The  curve  is 
concave  upward  on  one  side  of  a  point  of  inflection  and  concave 
downward  on  the  other  (see  Concavity). 

Not  every  root  of  Dxm  =  0  is  a  point  of  inflection.  It  is 
necessary  that  as  x  increases  through  the  root  in  question 
Dxm  shall  change  sign  (see  Maxima  and  Minima).  For  ex- 
ample, if  2/  =  x^,  then  m  =  Dxy  =  4a;^  and  Dxm  =  Vlx"^;  Dxm  =  0 
if  a;  =  0,  but  it  does  not  change  sign  as  x  increases  through 
zero.  Hence  the  origin  is  not  a  point  of  inflection.  It  is,  in 
fact,  a  minimum  point,  as  is  apparent  from  the  graph  (page  112), 


274 


ELEMENTARY  FUNCTIONS 


or' from  the  fact  that  DxV  =  4x^  changes  sign,  from  negative  to 
positive,  as  x  increases  through  zero. 

Since  m  is  the  rate  of  change  of  y, 

At  a  point  of  inflection  on  the  graph  of  a  function  the  rate  of 
change  of  the  function  has  a  maximum  or  minimum  valu£. 

The  corresponding  properties  of  the  graph  of  a  function  and  the  deriva- 
tive considered  in  this  section  may  be  indicated  as  follows: 


Property  of  Graph 
Slope  of  tangent  line. 
Maximum  or  minimum  point. 
Graph  rises  or  falls  to  the  right. 
Point  of  inflection. 
Graph  concave  upward  or  down- 
ward. 


Property  of  Derivative 
Value  of  Dxy. 
D,y  =  0. 

Dxy  is  positive  or  negative. 
Derivative  of  m  =  Dxy  is  zero. 
Derivative  of  m  is   positive  or  nega- 
tive. 


This  table  supplements  the  table  of  corresponding  properties  of  a  func- 
tion and  its  graph  given  in  Section  15. 
Example.     If 

find  the  points  of  maxima,  minima,  and  inflection.     Plot  the  graphs  of  y, 
m  =  Dxy  and  Dxm,  and  discuss  their  relations  to  each  other. 


Differentiating  (1)  m  =  Dxy 
and  differentiating  (2)      Dxm 

Setting  Dxy  =  0,  ^x' 

and  solving  for  x,  x 

Since  Dxm  =  6(2a;2  +  x-l) 
the  roots  of  Dxm  =  0  are       x 


=  4rc3  +  3x2  _  6^^ 
=  12x2  +  6x  -  6. 
+  3x2  _  6x  =  0, 
=  0,  0.9,  -  1.7. 
=  6(2x-l)(x  +  l), 
=  I  =  0.5,  and  -  1. 


(2) 
(3) 
(4) 
(5) 
(6) 
(7) 


y 

Dxy 

Dxm 

4 

0 

-6 

2.9 

0 

9.1 

-1.2 

0 

18.5 

3.4 

-1.8 

0 

1 

5 

0 

31 

-69 

84 

0 

-8 

30 

3 

1 

12 

16 

32 

54 

The  values  of  y,  Dxy,  and  Dxm  cor- 
responding to  the  values  of  x  in  (5)  and 
(7),  which  are  found  by  substitution  in 
(1),  (2),  and  (3),  are  given  in  the  table. 

The  point  A  (0,  4)  is  a  maximum  point 
since  the  tangent  line  is  horizontal 
iPxy  =  0)  and  the  curve  is  concave 
downward  (Dxi/<0).  While  the  points 
B(0.9,  2.9)  and  C(-  1.7,  -  1.2)  are  mini- 
mum points,  because  the  tangent  line 
is  horizontal  {Dxy  =  0)  and  the  curve  is 
concave  upward  (Z>xm>0). 

As  X  increases  through  0.5  or  -  1,  in- 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    275 

spection  of  the  changes  in  sign  of  the  factors  of  (6)  shows  that  D^m  changes 
sign  in  each  case.  Hence  the  points  D(0.5,  3.4)  and  E{-  1,  1)  are  both 
points  of  inflection. 

rhe  table  of  values  is  then  extended  to  include  the  necessary  integral 
values  (Section  50,  page  137),  the  computation  of  the  entire  table  being 
made  by  means  of  synthetic  division  and  the  remainder  theorem.  The 
graphs  are  then  drawn. 

The  maximum  and  minimum  points  oi  y,  A,  B,  C,  are  directly  above  or 
below  the  points  O,  B\  C,  at  which  DxP  cuts  the  x-axis.    The  vertical 


v>^ 

-V 

D. 

m 

-ft         -■^* 

"  iv^ 

\ 

^  t  t 

\ 

\ 

tit 

\ 

\ 

^ 

I  t-i 

\ 

\ 

t  ti 

\ 

z    i 

4^1 

\ 

\ 

e' 

6                       h 

1  / 

\ 

_J_ 

\ 

J.^       4 

17 

\ 

f 

y 

K 

\      ^ 

4 

\ 

1 

/ 

\ 

.      V 

T" 

\ 

7_ 

%     t- 

^r 

\ 

_d\ 

^  -T- 

ii 

■1 

\A~y 

'  '\ 

^^^' 

TiZ       J 

i 

' 

^c   ~ 

_ 

j[ 

1^ 

T 

\ 

i  / 

r 

\ 

*~t 

\~ 

\ 

r 

\ 

-^ 

1 

i>.v 

1 

Fig.  161. 


le  through  the  maximum  point  A  cuts  the  graph  of  Dxin  below  the  x-axis 
and  the  vertical  hnes  through  the  minimum  points  B  and  C  cut  it  above  the 
X-axis. 

The  points  of  inflection  of  y,  D  and  E,  the  minimum  and  maximum 
points  of  Dxy,  D'  and  E\  and  the  points  at  which  Dxtn  cuts  the  x-axis, 
D"  and  E",  are  respectively  on  the  same  vertical  hnes. 

The  graph  of  y  rises  (from  C  to  ^,  and  from  B  to  the  right)  or  falls 
(from  the  left  to  C  and  from  A  to  B)  according  as  the  graph  of  Dxy  lies 
above  or  below  the  a;-axis.  And  the  graph  of  Dxy  =  m  rises  or  falls  ac- 
cording as  that  of  D^m  hes  above  or  below  the  x-axis. 


276  ELEMENTARY  FUNCTIONS 

The  graph  of  y  is  concave  upward  (from  the  left  to  E  and  from  D  t6 
the  right)  or  downward  (from  E  to  D)  according  as  the  graph  of  Dxm  lies 
above  or  below  the  x-axis. 

What  further  relations  exist  between  the  graphs  of  Dxy  and  D^m? 

97.  Velocity  and  Acceleration.  The  velocity  of  a  body 
moving  along  a  line  is  the  rate  of  change  with  respect  to  the 
time  of  its  distance  s  measured  along  the  line  from  a  certain 
station  (Section  32,  page  93).     Hence, 

V  =  Dts,  (1) 

The  acceleration  a  of  a  moving  body  is  defined  to  be  the  rate 
of  change  of  its  velocity  v,  and  therefore 

a  =  Dtv.  (2) 

Example.  The  position  of  a  body  moving  on  a  line  at  the  tune  t  is 
given  by 

s  =  <3  -  5f2  +  2«  +  8.  (3) 

Find  the  velocity  and  acceleration  at  any  time.  Determine  the  posi- 
tion, velocity,  and  acceleration  when  i  =  -  1.     When  is  the  body  at  rest? 

Differentiating,  we  get 

V  =  Dts  =  3^2  _  lof  +  2,  (4) 

and  a  =  DtV  =  U  -  10.  (5) 

When  «  =  -  1,  we  find  that  s  =  Q,  v  =  lb,  and  a  =  -  16.  Hence  at  that 
time  the  body  passed  through  the  station,  moving  in  the  positive  direction 
at  the  rate  of  15  feet  per  second,  which  was  decreasing  at  the  rate  of  16 
feet  per  second  per  second. 

The  body  will  be  at  rest  at  any  instant  at  which  »  =  0;  that  is  when 
3^2  -  lOi  +  2  =  0.  (6) 

Solving  for  t, 

t  =  ?_±v^  =  0. 2  or  3. 2  seconds. 


EXERCISES 

'    1.  Differentiate  the  functions: 

(a)  3x^-20^-  4x2  +  5^  -  3.  (b)  ^  -Sx^  +  2x  -  5. 

(c)  3^ix^  +  5x  +  6).  (d)  {x  -I)  (x^  +  x-h  1). 

/  N  x3  -  7x2  +  4  a;  -  2     X  +  1 

(e)   ^ (f)  ^-X-^- 

2.  Differentiate  2/  =  (x2  -  3)2;    (a)  by  applying  (5),  Section  94,  regarding 
tt  =  x2  -  3;  (6)  by  first  removing  the  parentheses. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    277 

3.  Differentiate  the  functions  : 

(a)  2/  =  (a;2  +  2x)\  (6)  y  =  {2x^  -Sx  +  7)^ 

(C)    ?/  =  (X2  -  4)5,  {d)    2/  =  (3X3  -  5X  -  3)3. 

4.  Find  the  slope  of  the  line  tangent  to  the  graph  oi  y  =  x^  -  4:X  at  any 
point  P{x,  y);  at  the  point  for  which  x  =  1.  Plot  the  graph,  and  draw 
the  tangent  at  the  latter  point. 

5.  The  distance  fallen  in  t  seconds  by  a  ball  thrown  downward  with  a 
velocity  of  48  feet  per  second  is  s  =  48<  +  16^2  Yind  the  velocity  at  any 
time.     How  fast  will  it  be  moving  when  t  =  S? 

6.  A  billiard  ball  rolhng  down  a  smooth  plane  inclined  at  a  httle  less 
thaa  half  a  degree  moves  according  to  the  law 

Find  the  velocity  at  any  time.  Plot  the  graphs  of  s  and  v  on  the  same 
axes,  using  a  large  scale,  from  t  =  0  to  t  =  5.  From  the  graphs  answer  the 
following : 

(a)  What  is  the  position  and  velocity  of  the  ball  after  4.5  seconds? 

(b)  With  what  velocity  will  the  ball  be  moving  after  it  has  rolled  3  feet? 

(c)  How  far  must  it  roll  to  acquire  a  velocity  of  2  feet  per  second? 

(d)  When  is  the  distance  equal  to  the  velocity?     How  many  solutions? 

(e)  Draw  a  tangent  to  the  graph  of  s  and  measure  its  slope.  Measure 
the  ordinate  on  the  point  on  the  graph  of  v  which  has  the  same  abscissa 
as  the  point  of  tangency.     How  do  the  two  compare? 

7.  Find  the  points  at  which  the  tangents  to  the  graph  of  ^  =  a:^  -  27a; 
are  horizontal.     Construct  the  figure. 

8.  The  height  after  t  seconds  of  a  body  thrown  vertically  upward  with 
a  velocity  of  96  feet  per  second  is  s  =  96^  -  16i-.  Find  the  velocity  at 
any  time.     When  will  the  velocity  be  zero?    How  high  will  the  body  rise? 

9.  Find  the  angles  which  the  graph  of  y  =  —  x^  +  5x^  —  Qx  makes  with 
the  X-axis. 

10.  At  what  angles  does  the  line  y  =  4x  cut  the  graph  oi  y  =  x^? 

11.  Oil  dropped  on  the  floor  spreads  out  in  a  circle.  Find  the  rate  at 
which  the  circumference  increases  with  respect  to  the  radius,  and  the  rate 
at  which  the  area  increases  with  respect  to  the  radius. 

12.  The  kinetic  energy  of  a  body  of  mass  m  moving  with  a  velocity  v  is 
given  by 

K  =  |m»2. 

Find  the  rate  at  which  the  kinetic  energy  changes  with  respect  to  the 
velocity. 

13.  If  P  is  the  pressure  of  a  body  on  a  surface  and  F  the  friction  between 
them,  what  does  the  derivative  of  F  with  respect  to  P  represent? 

14.  If  s  is  a  quadratic  function  of  t,  show  that  the  acceleration  is  uni- 
form (constant). 


278  ELEMENTARY  FUNCTIONS 

15.  Find  the  points  of  maxima,  minima,  and  inflection  on  the  graphj 
of  each  of  the  equations  below.     On  the  same  axes,  plot  the  graphs  of 
m  =  Dxy,  and  DxW,  and  discuss  the  relations  between  them. 

(a)  y  =^  x^  -  4x  +  5.  (h)  y  =  x^  +  2x^  -  4x  -  3. 

(c)  y^x^-  Qx^  +  \2x  +  3.  {d)  y  =  a^  ■\-  Zx^  +  5z  -  2. 

(e)  2/  =  3a:*  -  4a;3  _  63^2  +  12a;  -  2.     (f)    y  =^  x^  -  x^  -2x +  Zx  -  1. 
(g)  y  =  x^  -bx^  +  2x  +  8. 

16.  The  distance  from  the  starting  point,  after  t  seconds,  to  a  ball  rollec 
up  a  plane  inchned  at  a  Uttle  less  than  half  a  degree  with  an  initial  velocitj 
of  8  feet  per  second  is 

How  far  will  it  roll  up  the  plane,  and  how  fast  will  it  be  moving  when  i\ 
returns  to  the  starting  point? 

17.  The  position  of  a  body  moving  on  a  line  is  given  by  one  of  the 
equations  following.  Find  the  velocity  and  acceleration  at  any  time.  Oi 
the  same  axes  plot  the  graphs  of  s,  »,  and  a,  and  discuss  the  motion  witl 
reference  to  each  of  them. 

(a)  s  =  f-U-\-4..  (b)  s  =  <3  -  6^2  +  i2t  _  8. 

(c)  s  =  <3  +  ^2  _  6f,  (d)  s  =  2^  -  8^3  -  9^2  +  54i. 

18.  Find  the  largest  possible  number  of  horizontal  tangents  to  the  grapl 
of  a  polynomial  of  degree  n;  the  largest  possible  number  of  points  of  ia 
flection. 

19.  If  the  graph  of  a  function  y  is  concave  toward  the  a;-axis,  show  tha 
y  and  D^m  have  opposite  signs. 

20.  li  y  =  ax^  +  dx"^  +  ex  +  d,  show  that  the  abscissas  of  the  points  oi 
the  graph  at  which  the  tangent  line  is  horizontal  satisfy  a  quadratic  equa 
tion.  Find  the  condition  that  the  number  of  horizontal  tangents  is  two 
one,  or  zero.  Apply  this  condition  to  determine  the  number  of  horizonta 
tangents  in  15,  6,  c,  d. 

21.  li  y  =  uv,  show  that  Dxy  =  uDxV  +  vDxU. 

98.  Derivative  of  a  Rational  Function.  In  order  to  dif 
ferentiate  any  rational  function  we  need  but  one  more  rule. 

Theorem.     The  derivative  of  a  fraction  is  a  fraction  whoSi 

numerator  is  the  denominator  times  the  derivative  of  the  numerata 

less  the  numerator  tim^s  the  derivative  of  the  denominator,  am 

whose  denominator  is  the  square  of  the  denominator.    Symbolically 

p.  u  _  vDxU  -  uDzV 

v  v^ 

T       X  ^ 

Let  2/  =  7 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    279 
len 


u  -\-  Au 


whence 
Ay  = 
Hence 


u  +  Au 
V  +  Av 


'  '  ^'-  v  +  Av' 

u      uv  +  vAu  -  uv  - 

uAv 

vAu  —  uAv 

V                v{v  +  Av) 

v{v  +  Av) 

Au        Ay 
Ay      '^Ax'^'Ax 
Ax        v(v  +  Av) ' 

Finding  the  limit  as  Ax  approaches  zero,  we  get 

vDxU  —  uDxV 


Example  1. 


-(fel) 


D.y  = 

I/- 

_  (x^  +  5)  D,(Sx  -  4)  -  (3a;  -  4)  D:,(x^  +  5) 

{x^  +  5)2 

(x^  +  5)  (3)  -  (3a;  -  4)  (2x) 

(x2  +  5)2 
-  3x2  +  ga;  +  15 


(a;2  +  5)2 

After  one  fixes  the  rule  in  mind,  it  becomes  easy  to  write  down  the  second 
^  fraction  on  the  right  without  taking  the  time  to  write  out  the  first. 

Example  2.     Find  the  derivative  of  y  =  1/w",  where  it  is  a  function  of  x. 

1      w"  DA  -  1  Dxw« 


re  have 


D. 


w* 


0  -  nu'^~^DxU 
—  nDxU 


by  (1)  and  (5),  Section  94. 


Since  the  result  obtained  in  Example  2  may  be  written  in  the 

Dxiu-"")  =  -  nir^-^DxU, 

see  that  Theorem  5,  Section  94,  holds  for  negative  as  well  as 
positive,  integral  values  of  the  exponent  n. 

99.  Derivative  of  an  Irrational  Function.  In  order  to  dif- 
ferentiate an  irrational  function  it  is  sufficient  to  show  that 
Theorem  5,  Section  94,  holds  also  for  a  fractional  exponent, 
n  =  p/q. 


280  ELEMENTARY  FUNCTIONS 

Let  y  =  wP/«  (1) 

Raising  both  sides  to  the  qth  power, 

2/«  =  w^.  (2) 

As  2/«  and  u^  are  merely  different  notations  for  the  same 
function  of  x,  their  derivatives  are  the  same.    Hence 

L  D.r  =  D,u^'  (33 

Applying  Theorem  5,  Section  94, 

qy^-^DxV  =  puP-^DxU,  (4 

Dividing  (4)  by(2), 

qy-^D^y  =  pu-^DxU.  (6| 

Multiplying  (5)  by  (1),  and  dividing  by  q, 

V    -  —1 
Dxy  =  -UQ     DxU 

=  nU^~^DxU,    where  n  =  -• 

Q 
Hence  we  have  the 

Theorem.    If  y  =  W^,  where  in  is  a  fraction^  then  Dxy  =  nW'^DxU, 

Example.     Differentiate  y  =  Vx^  -  a?. 

Since  y  =  {x^  -  a2)l, 

and  since  we  may  regard  x^  -  a^  as  a  function  u,  we  have,  by  the  theorem, 

J>*y  =  Ka:^  -  a')-i2x. 

X 


Vx^  -  a2 


EXERCISES 

1.  Find  the  derivatives  of  the  following  fimctions: 

(^)  ^^-  (^^  3^34-  (^)   ^• 

^^^iiTs'  ^"^^m:-!-  (^)^^32i- 


(g)  Va:2  _  2a;.  (h)  Vl  -  xK  (i)  Vx. 


(j)   a;V2x  -  3.      Hint:  xV2x  -  3  =  V2x»  -  3x2.     (k)  aj^-j-z ._  5^;. 
(1)   x^{2x  +  4)|.  (m)  -=4==-  (^)  (^'  +  3x  -  5)1 

^"^  ^^%^-  <P^  »»V2j^rS  (q)  ^^- 


i 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    281 


2.  If  the  temperature  of  a  gas  is  constant,  the  pressure  is  inversely- 
proportional  to  the  volume.  Find  the  rate  of  change  of  the  pressure  with 
respect  to  the  volume. 

3.  The  intensity  of  light  varies  inversely  as  the  square  of  the  distance 
from  the  source.  Find  the  ratio  of  the  intensities  at  two  points,  one  2 
feet  from  the  source,  the  other  4  feet.  If  a  body  moves  away  from  the 
source,  find  the  rate  at  which  the  intensity  changes  as  the  distance  increases. 
Compare  the  rates  of  change  (find  their  ratio)  at  the  two  points  given 
above. 

4.  Find  the  points  at  which  tangents  to  the  graphs  of  the  following 
equations  are  horizontal,  if  any,  and  construct  the  figures. 

(a)  y  =  z — o  (b)  y  =  -X — ;:  (c)  y 


a:2  +  l 

5.  Construct  the  graph  of  x^  -  xy  ^  4,  and  find  the  points  at  which  the 
tangent  hne  is  horizontal.     Has  the  curve  any  points  of  inflection? 

6.  Find  the  points  of  inflection  of  the  graph  of  ?/  =  x/Cx^  +  1). 

7.  The  original  amount  of  carbon  monoxide  in  a  mixture  of  formic  and 
sulphuric  acids  is  a,  and  the  amount  x  produced  in  the  time  i  is  given  by 
the  equation 


H,  where  ^'  is  a  constant. 


t 


a{a  -  x) 

Find  the  rate  at  which  carbon  monoxide  is  formed. 

8.  Find  the  angle  at  which  the  graphs  of  ?/  =  l/x^  and  y  =  :ir  intersect. 

9.  Find  the  distance  from  the  origin  to  the  point  (x,  y)  in  terms  of  x 
and  y,  and  show  that  the  graph  of  x"^  ^-y"^  =  16  is  a  circle.  Find  the  slope 
of  the  tangent  line  at  any  point. 

10.  Show  that  the  graph  of  y  =  +V3^  is  concave  upward,  and  that  of 
y  =  -Vx^  is  concave  downward.  Show  that  both  graphs  are  tangent  to 
the  a:-axis  at  the  origin.  What,  then,  is  the  form  of  y  =  x^  =  =tV5^  near 
the  origin? 

100.  Equations  of  Tangent  and  Normal  Lines.  The  slope  of 
the  Hne  tangent  to  a  curve  at  any  point  Pi(xi,  i/i)  on  it,  is  the 
value  of  the  derivative  at  Pi,  that  is,  the  value  of  Dxy  for  x  =  Xi. 
Hence  the  equation  of  the  line  tangent  at  Pi  may  be  found  by 
the  equation  y  -  yi  =  m{x  -  X\) ,  given  on  page  66. 

Definition.  The  normal  to  the  curve  at  any  point  on  the 
curve  is  the  line  perpendicular  to  the  tangent  at  that  point. 

If  m  is  the  slope  of  the  tangent  hne,  then  the  slope  of  the 
normal  is  -  1/m,  since  the  slope  of  one  of  two  perpendicular 
lines  is  the  negative  reciprocal  of  the  slope  of  the  other  (page 
200). 


282 

Example  1 
of 


ELEMENTARY  FUNCTIONS 
Find  the  equation  of  the  tangent  and  normal  to  the  graph 

at  the  point  for  which  x  =  4. 

The  slope  of  the  tangent  Une  at  any  point  is  m  =  Dxy  =  \x,  and  hence  that 

of  the  normal  is 

1         2 

m  = =  _  -. 

m         X 

At  the  point  Pi  for  which  a:i  =  4,  we 
have  2/1  =  4,  m  =  2.  Hence  the  equa- 
tion of  the  tangent  is 

2/  -  4  =  2(a:  -  4) 
or  2a;  -  2/  -  4  =  0.  (1) 

At  Pi  (4,  4)  the  slope  of  the  normal 
is  m'  =  -  \.  Hence  the  equation  ol 
the  normal  is 

4  =  -  Ux  -  4) 


X      4d-      -L 

X4^    .V     7 

"^     :  ^^^ 

I    ^'''"'  j^ 

^^  1    -,t 

^        4 

^--^' 

-6    ■       -3    -i     -1    0       1   J/S     3      i      i      6   i 

Z    T 

-U- 

T       -V 

jj^          i 

iFiG.  162. 


y 


or 


X  +  2y  -  12  =  0.  (2) 

Example  2.  Find  the  equation  of  the  tangent  and  normal  to  the 
parabola  in  Example  1  at  any  point  |Pi(a:i,  i/i)  on  the  curve.  If  M  ia 
the  projection  of  Pi  on  the  2/-axis,  and  if  the  tangent  and  normal  cut 
the  ?/-axis  at  T  and  iV  respectively,  show  that  0  bisects  TM  and  that 
AfiV  is  constant. 

At  Pi  the  slope  of  the  tangent  is  m  =  a;i/2,  and  hence  the  equation  of 
the  tangent  is 

xi).  .  (3) 


y 


Xi  . 
2/1  =  2  (^ 


Setting  a:  =  0  in  (3)  and  solving  for  y,  the  intercept  on  the  y-axis  is 

y  =  yi--^'  (4) 


Since  Pi  lies  on  the  graph  oi  y  =  x^/4:,  its  coordinates  satisfy  this  equation 
80  that  2/1  =  Xi^/4^  and  hence  xi^  =  4yi.    Substituting  in  (4) 


2/  =  2/i  - 


4vi 


2/1. 


Hence  OT  =  -  i/i,  and  as  OM  =  yi,  we  have  OT  =  -  OM,  so  that  0  is 
the  middle  point  of  TM. 

[    The  tangent  at  any  point  Pi  may  therefore  be  constructed  by  laying 
off  on  the  2/-axis  OT  =  -  2/1,  and  joining  T  to  Pi. 

The  slope  of  the  normal  at  Pi  is  m  -  -  2/ari.     Hence  the  equation 
the  normal  is 

2 


2/1  » 


Xi 


(x  -  Xi). 


i 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    283 

The  intercept  of  (5)  on  the  2/-axis  is 

2/  =  2/1  +  2. 

Hence  N  lies  2  units  above  Pi,  or  MN  =  2,  a  constant. 
The  normal  at  Pi  may  be  constructed  by  drawing  the  line  through  Pi 
i  and  the  point  N  which  is  2  units  above  the  projection  of  Pi  on  the  y-&xia. 

A  frequent  source  of  error  in  finding  the  equation  of  the 
tangent  or  normal  to  a  curve  at  any  point  Pi  on  the  curve  is 
the  failure  to  keep  in  mind  that  x  and  y  are  variables,  the 
coordinates  of  any  point  on  the  Une,  while  Xi  and  2/1  are  con- 
stants, the  coordinates  of  a  fixed  point.  The  slope  of  the  tangent 
or  normal  Une  at  a  given  point  Fi  is  a  constant. 


EXERCISES 

1.  Obtain  equations  (1)  and  (2)  above  by  substituting  the  coordinates 
of  Pi  (4,  4)  in  equations  (3)  and  (5) . 

2.  Show  that  the  length  of  the  line  joining  Pi(a:i,  ^1)  and  P2{x2,  yt) 


is  ^/{Xl  -  X2Y  + 


y^y 


3.  Show  that  the  coordinates  of  the  middle  point  of  the  line  P1P2  are 
X  =  (xi +a:2)/2,  y  =  (yi  +  y2)/2.  Hint:  If  P  is  the  middle  point,  the 
values  of  Ax  computed  for  Pi  and  P  and  for  P  and  P2  are  equal;  and 
so  also  are  the  values  of  Ay. 

4.  Find  the  equations  of  the  tangent  and  normal  to  the  following  curves 
at  the  points  indicated.    Construct  the  figure  in  each  case. 

(a)  y  =  x^-4x,  (3,  -  3). 

(b)  2/  =  3  +  2x-x2,  (1,4). 

(c)  2/ =  0:3^  (1,1). 

(d)  y  =  a^  -4x  +  S,  point  of  inflection. 

(e)  xy  =  4,  (4,  1). 

(f)  xy-y  =  2x,  (2,4). 

6.  Find  the  equations  of  the  tangent  and 
nonnal  to  the  parabola  2/  =  a;^  at  any  point 
Piixi,  yi)  on  the  parabola. 

6.  Find  the  equations  of  the  tangent  and  normal  to  the  hyperbola 
xy  =  4  at  any  point  Pi  (2:1,  yi). 

7.  Find  the  equations  of  the  tangent  and  normal  to  the  parabola  y  =  ax^ 
at  any  point  Pi(a;i,  yi).  If  these  lines  cut  the  i/-axis  at  T  and  N  respect- 
ively, and  if  M  is  the  projection  of  Pi  on  the  y-axis,  show  that  the  origin 
0  bisects  TM,  and  that  MN  is  constant.  State  a  rule  for  constructing  the 
tangent  and  normal  at  any  point  on  the  parabola. 


Fig.  163. 


284 


ELEMENTARY  FUNCTIONS 


8.  Find  the  coordinates  of  the  point  at  which  the  tangent  to  the  para- 
bola y  =  ax^  at  Pi  cuts  the  x-axis.  If  the  tangent  cuts  the  a:-axis  at  72, 
and  the  ?/-axis  at  T,  prove  that  R  is  the  middle  point  of  PiT.  Find  the 
equation  of  the  hne  through  R  perpendicular  to  the  tangent.  Show  that 
this  line  always  cuts  the  y-axis  at  the  same  point  F,  no  matter  what  point 
on  the  curve  Pi  is. 

Definition.  The  point  P(0,  l/4o)  is  called  the  focm  of  the  parabola 
y  =  ax"^.  The  focus  is  the  point  at  which  the  line  in  Exercise  8  cuts  the 
^-axis. 

9.  In  the  figure,  PiA  is  parallel  to  the  y-axis.  By  means  of  Exercise  8, 
and  methods  of  plane  geometry,  show  that  angle  FPiN  =  angle  NPiA^ 

the  notation  being  the  same  as  ii 
Exercises  7  and  8. 

Note  :  A  parabolic  reflector,  sue! 
as  is  used  in  a  headlight  of  an  auto 
mobile,  is  formed  by  revolving 
parabola  about  its  axis  of  sym 
metry.  If  a  source  of  light  is  placec 
at  P,  Exercise  9  shows  that  th< 
rays  will  be  reflected  in  paralle 
lines. 

10.  Find  the  equations  of  tb 
lines  tangent  to  the  parabola  y  =  ax 
at  the  two  points  for  which  y  =  l/4a 
Where  do  they  intersect?  At  whai 
angle? 

11.  Find  the  equation  of  the  lin( 
through  the  focus  P(0,  l/4a)  per- 
pendicular to  the  tangent  to  th< 

parabola  at  Pi(xi,  yi).    Show  that  it  intersects  the  line  a;  =  xi  on  the  lin< 
2/  =  -  l/4a. 

Definition.  The  line  y  =  -  l/4a  is  called  the  directrix  of  the  paraboll 
y  =  ax^. 

12.  Given  the  parabola  y  =  ax^  and  a  point  Pi(a;i,  j/i)  on  it,  show  thai 
the  distance  from  the  focus  F(0,  l/4a)  to  Pi  is  PPi  =  t/i  +  l/4a.  Shov 
from  this  that  any  point  on  a  parabola  is  equidistant  from  the  focus  anc 
the  directrix. 

13.  Find  the  equation  of  the  tangent  to  the  graph  oi  y  =  ax^  at  any 
point  Pi.  If  it  cuts  the  y-Sixia  at  T,  and  if  M  is  the  projection  of  Pi  on 
the  2/-axis,  show  that  the  origin  trisects  TM. 

14.  Find  the  equation  of  the  Hne  tangent  to  the  parabola  y^  =  x  at  any 
point  Pi  on  the  curve.  Show  that  its  intercept  on  the  y-axis  is  half  the 
ordinate  of  the  point  of  contact  and  that  it  is  perpendicular  to  the  line 
joining  the  point  of  intersection  with  the  y-axis  to  the  focus. 


y 

J 

'/ 

r 

\        ^ 

/7 

I 

\f< 

w 

0 

r 

X 

'1 

1 

Fig.  164. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    285 

15.  Let  F  be  a  fixed  point  on  the  a:-axis  and  T  any  point  on  the  2/-axis. 
Through  T  draw  the  hne  perpendicular  to  TF.  Choosing  different  positions 
for  T,  draw  a  number  of  such  lines,  enough  so  that  the  form  of  the  parabola 
to  which  they  are  tangent  becomes  apparent. 

16.  If  Pi  is  a  point  on  the  equilateral  hyperbola  xy  ='  a,  M  its  projec- 
tion on  the  y-axis,  and  if  the  tangent  Une  cuts  the  y-axis  at  T,  then  M  is 
the  middle  point  of  OT.    How  can  this  be  used  to  construct  the  line  tan- 

1  gent  at  a  given  point  on  the  curve? 

17.  Show  that  the  point  of  contact  of  a  line  tangent  to  the  equilateral  hy- 
I  perbola  xy  =  ais  the  middle  point  of  the  segment  included  between  the  axes. 

f  18.  Find  the  area  included  between  the  axes  and  the  hne  tangent  to 
1  the  equilateral  hyperbola  xy  ^^  a  at  any  point  Pi.  State  the  result  as  a 
,   theorem. 

S  19.  If  a  normal  is  drawn  to  the  equilateral  hyperbola  xy  =  a  at  a  point 
|i  Pi  on  it,  then  Pi  is  the  middle  point  of  the  segment  included  between  the 
\  Imes  bisecting  the  coordinate  axes. 


i      101.  Problems  in  Maxima  and  Minima.    To  solve  a  problem 
11  involving  maximum  and  minimum  values,  it  is  necessary  first 
'  to  express  the  variable  v  which  is  to  be  a  maximum  or  mini- 
mum in  terms  of  a  single  independent  variable  x.     The  quantity 
which  is  to  be  made  a  maximum  or  minimum  is  usually  appar- 
ent from  the  statement  of  the  problem,  but  there  is  frequently 
some  choice  in  the  selection  of  the  independent  variable.     In 
I  many  problems,  as  in  Example  2  below,  the  variable  v  may  be 
=  expressed  at  once  in  terms  of  two  variables,  one  of  which  may 
I  be  eliminated  by  means  of  a  relation  between  them.     Having 
I  found   the  function   v  =  f(x),   one  then  differentiates  v  with 
r  respect  to  x,  sets  the  derivative  equal  to  zero,  and  solves  for  x. 
I  It  must  then  be  shown  that  at  least  one  of 
J  the  values  of  x  so  found  makes  v  a  maximum 
or  minimum,  and  this  maximum  or  minimum 
!    value  can  be  determined  by  substitution  in 
'    v^f{x). 

j  Example    1.     A  box  is  to  be  made  out  of  a 

I  square  piece  of  cardboard,  four  inches  on  a  side,  by              p^^   jgg 

j  cutting  out  equal  squares  from  the  corners  and 

I  then  turning  up  the  sides.     Find  the  dimensions  of  the  largest  box  that 

t  can  be  made  in  this  way. 

j  If  the  squares  cut  out  are  small,  the  box  will  have  a  large  base  and  a 

'  shallow  depth.    If  the  squares  are  large,  the  base  will  be  small  and  the 


<-4-2a?— » 

«-a;-> 

"*-(r-» 

286 


ELEMENTARY  FUNCTIONS 


box  deep.  In  either  case  the  volume  will  be  small.  Somewhere  between 
these  two  extremes  will  be  a  box  whose  volume  is  greater  than  that  of  any 
other,  that  is,  a  box  of  maximum  volume. 

Let  X  be  the  side  of  the  square  cut  out.  Then  the  depth  of  the  box 
X,  and  the  side  of  the  base  is  4  -  2x.  Hence  the  volume,  expressed  as 
function  of  a;  is 

F  =  x(4  -  2x)2  =  16x  -  16x2  +  ^3?. 

The  graph  of  F  is  readily  plotted.  From  it,  the  value  of  x  which  makej 
F  a  maximum  appears  to  be  somewhat  less  than  unity.  By  computing 
F  for  a  large  number  of  values  of  x,  we  could 
approximate  the  best  value  of  x.  By  means  of  th( 
derivative  we  can  avoid  this  labor,  and  obtain  thei 
exact  value. 

At    the    maximimi  point    the   tangent    line 
horizontal,   and  hence  its  slope  is  zero,  so  that 
DxF  =  0. 

We  therefore  compute  the  derivative,  set  it 
equal  to  zero,  and  find  the  value  of  x  which  pro^ 
duces  this  result.    We  then  have 


-.1            T 

4-      t 

^^p-     t 

l-^  J~ 

S  .^1   ^ 

ii.txl? 

Fig.  166. 


or,  dividing  by  4, 

Factoring 
and  hence 


DxF  =  16  -  Z2x  +  12x2  ^  0 

3x2  -  8x  +  4  =  0. 

(3x  -  2)  (x  -  2)  =  0, 

X  =  2/3  or  2. 


From  the  figure,  we  see  that  x  =  f  makes  F  a  maximum,  and  x  = 
gives  a  minimum.  That  x  =  2  gives  a  minimum  follows  also  from  the  fact 
that  if  X  =  2,  then  F  =  0,  that  is,  the  box  will  not  hold  anjd^hing.  The 
preliminary  discussion  showed  the  existence  of  a  maximum,  and  as  x  =  ^ 
is  the  only  other  possibility  it  must  be  the  value  of  x  for  which  F  is  a  maxi- 
mum. Either  of  the  criteria  in  Section  96  may  also  be  appUed  to  show  thati 
F  is  a  maximum  if  x  =  f . 

In  order  then,  to  have  a  box  of  maximum  capacity,  we  must  cut  oul 
squares  from  the  corners  f  of  an  inch  on  a  side.    The  depth  of  the  box 
will  be  I  of  an  inch,  the  side  of  the  base  will  be  4  -  2  x  f  =  2|  inches,^ 
and  the  capacity  will  be  F  =  (2f)2  x  f  =  ^^^  cu.  in. 

Example  2.  As  large  a  rectangular  stick  of  timber  as  possible  is  to  be 
sawed  from  a  log  10  inches  in  diameter  at  the  smaller  end,  the  length  of 
the  stick  to  be  the  same  as  that  of  the  log.     Find  its  other  dimensions. 

Let  F  denote  the  volume  of  the  stick  of  timber,  I  its  length,  and  A  the 
area  of  an  end.  Then  F  =  Z^l,  and  since  I  is  constant,  the  volume  F  will 
be  a  maximum  if  and  only  if  A  is  a  maximum.  The  end  of  the  stick  is  a 
rectangle  inscribed  in  a  circle  whose  diameter  is  10,  and  the  problem  re- 
duces to  a  determination  of  the  dimensions  of  the  maximum  rectangle 
which  can  be  inscribed  in  this  circle. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    287 


The  area  of  the  rectangle  is 

A  =  xy,  (1) 

rhich  involves  the  two  variable  dimensions  x  and  y.    These  are  connected 
)y  the  relation  x"^  +  y^  =  100. 
Solving  for  y, 


y  =  V 100  -  x^ . 
and  substituting  in  (1) 

A  =  xVlOO  -  x\ 

In  order  to  find  the  derivative  of  A  we  write  it  in  the  form 

A  =  Vl00a;2  -  x*  =  (lOOx^-  x*)K 
lifferentiating, 

DxA  =  K100a;2  -  a:4)-j(200x  -  4^^) 
lOOx  -27^        100  -  2x2 


(2) 
(3) 


Vl00x2  -  X*     VlOO  -  x^ 


luating  the   derivative  to 


100  -  2x^ 


0. 


viOO  -x^ 

[ultiplying    both   sides   by 
ioo  -  x\ 

100  -  2x2  =  0, 
id  solving  for  x, 

x  =  ^  5  V2. 

The  negative  value  of  x  has 
'ho  meaning  in  this  problem. 
That  the  value  of  A  is  a  maxi- 
mum if  X  =  5  V2  may  be  seen 
as  follows:  If  x  is  very  small, 
or  very  near  10,  A  is  very  Pj^    j^gy 

small,  while  larger  rectangles 

lie  between  these  extremes.  Hence  as  x  increases  from  0  to  10,  A  first 
increases  and  then  decreases,  and  as  A  does  not  become  infinite  it  therefore 
has  a  maximum  in  this  interval.  But  x  =  5  V2  is  the  only  point  in  this 
interval  at  which  the  tangent  to  the  graph  of  A  is  horizontal,  and  it  must 
therefore  give  the  maximum  value.  _ 

The  corresponding  value  of  y,  from  (2),  is  ?/  =  5  V2.  Hencethe  end  of 
the  largest  stick  of  timber  will  be  a  square  whose  side  is  5  V2  inches,  or 
very  nearly  7  inches. 


288  ELEMENTARY  FUNCTIONS 

EXERCISES 

1.  A  box  is  to  be  made  by  cutting  squares  from  the  comers  of  a  piece 
of  cardboard  6  by  8  inches,  and  folding  up  the  sides.  Find  the  dimensions 
if  the  capacity  is  to  be  a  maximum. 

2.  A  chicken  yard  is  to  be  made  from  36  feet  of  poultry  fencing,  the 
side  of  a  bam  being  used  for  one  side  of  the  yard.  Find  the  dimensions 
in  order  that  the  yard  may  be  as  large  as  possible. 

3.  Find  the  dimensions  and  capacity  of  the  largest  box  which  can  b< 
made  with  a  square  base  and  no  top  if  the  total  amount  of  cardboard  ii 
the  box  is  48  square  inches. 

4.  What  should  be  the  dimensions  of  a  rectangular  garden  plot  with 
perimeter  of  12  rods,  in  order  to  have  the  greatest  area  possible? 

5.  A  two  acre  pasture  in  the  form  of  a  rectangle  is  to  be  fenced  off  alonj 
the  bank  of  a  straight  river,  no  fence  being  needed  along  the  river.  Find 
the  dimensions,  in  rods,  in  order  that  the  fence  may  cost  as  little  as  possible 

6.  The  legs  of  an  isosceles  triangle  are  6  inches  long.  How  long  must 
the  base  be  in  order  that  the  area  may  be  a  maximum? 

7.  The  height  of  a  rifle  ball  fired  vertically  upward  with  an  initial 
velocity  of  1200  feet  per  second  is  s  =  1200«  -  16^^^     How  high  will  it  rise? 

8.  By  the  Parcel  Post  regulations,  the  combined  length  and  girth  of  9 
package  must  not  exceed  6  feet.  Find  the  dimensions  and  volume  of  the 
largest  parcel  which  can  be  sent  in  the  shape  of  a  box  with  square  ends. 

9.  A  farmer  has  150  rods  of  fencing.     Find  the  dimensions  and  area  ol 
the  largest  rectangular  field  he  can  enclose  and  divide  into  two  equal  pai 
by  a  fence  parallel  to  two  of  the  sides. 

10.  If  the  total  area  of  the  field  in  the  preceding  Exercise  is  to  be  1^ 
square  rods,  find  the  dimensions  and  the  amount  of  fencing  needed  if  the 
latter  is  to  be  a  minimimi. 

11.  A  rectangular  cistern  is  to  be  built  with  a  square  base  and  open  top./ 
Find  the  proportions  if  the  amount  of  material  used  is  to  be  a  minimum. 

12.  A  rectangular  piece  of  ground  is  to  be  fenced  off  and  divided  into 
three  equal  parts  by  fences  parallel  to  one  of  the  sides.  What  should  the 
dimensions  be  in  order  that  as  much  ground  as  possible  may  be  enclosed 
with  16  rods  of  fence? 

13.  If  the  total  area  enclosed  in  Exercise  12  is  an  acre,  find  the  dimensions 
in  order  that  the  total  length  of  the  fence  should  be  a  minimum. 

14.  The  number  of  tons  of  coal  consumed  per  hour  by  a  certain  ship  is 
0.3  +  0.001  V^,  where  V  is  the  speed  in  knots.  For  a  voyage  of  1000 
knots  at  V  knots  per  hour,  find  the  total  consumption  of  coal.  For  what 
speed  is  the  consumption  of  coal  least? 

15.  Divide  a  string  16  inches  long  into  two  parts,  so  that  the  combined 
area  of  the  square  and  circle  with  perimeters  equal  to  the  parts  shall  be  a 
minimum. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    289 


16.  Find  the  coordinates  of  the  maximum  or  minimum  point  of  ?/  -= 
I    ax^  +  bx  +  c. 

17.  At  what  height  should  a  light  be  placed  above  a  writing  table  in 
order  that  a  small  portion  of  the  table,  at  a  given  horizontal  distance  d 
from  the  point  directly  below  the  light,  may  receive  the  greatest  illumina- 

' '  tion  possible?  (It  is  known  that  the  intensity  of  illumination  varies  in- 
i  versely  as  the  square  of  the  distance  and  directly  as  the  sine  of  the  angle 
f  between  the  line  from  the  light  to  the  point  in  question  and  the  table. 
i  Express  the  illumination  as  a  function  of  the  height  of  the  light,  and  find 
I    the  value  of  the  height  which  gives  the  maximum  illumination.) 

18.  A  rectangular  box  is  to  be  made  by  cutting  out  the  comers  of  a 
I  rectangular  piece  of  cardboard  and  folding  up  the  sides.  If  the  depth  of 
;  the  box  is  to  be  2  inches,  find  the  dimensions  of  the  smallest  piece  of  card- 
I    board  which  will  make  a  box  to  contain  72  cubic  inches. 

I  102.  Related  Rates.  If  two  variables  x  and  y  are  functions 
I  of  the  time  t,  then  their  derivatives  with  respect  to  t,  DtX  and 
I  Dty,  measure  the  rates  at  which  x  and  y  are  changing.  These 
I  rates  will  be  related,  that  is,  connected  by  a  relation,  if  y  is 
I   a  function  of  x. 

!  For  example,  if  x  and  y  are  functions  of  t  such  that  y  =  x^ 
!  then  the  rates  at  which  x  and  y  change  satisfy  the  equation 
j  Dty  =  SxWtX,  which  is  obtained  by  differentiating  the  given 
i   equation  with  respect  to  t.    The  rate  of  change  of  y  depends 

on  the  value  of  x  as  well  as  on  that  of  the  rate  of  change  of  x. 
,  In  solving  problems  involving  the  rates  at  which  two  varia- 
|i  bles  change,  it  is  first  necessary  to  determine  the  relation  y  =  f{x) 
I  connecting  the  variables.    Then  the  relation  between  the  rates 

of  change  of  x  and  y  is  found  by  differentiating  y  =  f{x)  with 

respect  to  t. 

Example  1.    Oil  dropped  on  a  smooth  floor  spreads  out  in  the  form  of 
a  circle.     If  the  radius  is  increasing  at  the  rate  of  |  an  inch  per  second 
>  when  it  is  6  inches  long,  how  fast  is  the  area  increasing? 

If  r  denotes  the  radius  and  A  the  area,  the  question  asked  may  be  ex- 
[>  pressed  symbolically  as  follows:  If  Dtr  =  |  when  r  =  6,  what  is  the  value 
)  of  DtA'f 

In  order  to  answer  this  question  we  must  have  a  relation  between  DiA 
and  Dtr.  And  to  obtain  this  relation  we  must  first  express  .4  as  a  function 
of  r.    From  plane  geometry  we  have 

A  =  Trr^. 


290 


ELEMENTARY  FUNCTIONS 


Differentiating  both  sides  with  respect  to  t,  we  get 

DA  =  2TrDtr. 

Substituting  the  values  of  tt,  r,  and  Dtr,  the  oil  covered  area  is  increasing 
at  the  rate  of  DtA  =  2  x  22/7  x  6  x  |  =  18.85  square  inches  per  second. 

Example  2.     A  man  walks  along  a  sidewalk  at  the  rate  of  three  miles 
an  hour  (4.4  feet  per  second),  approaching  a  house  which  stands  back  7 
feet  from  the  walk.     When  he  is  24  feet  from  the 
walk  leading  to  the  house,  how  rapidly  is  he' 
approaching  the  house? 

The  rate  at  which  he  approaches  the  house  is 
the  derivative  of  his  distance  from  the  house 
with  respect  to  time.  If  y  denotes  his  distance 
from  the  house,  we  seek  Dty. 

The  rate  at  which  the  man  walks  is  the  rate 
of  change  of  his  distance  from  some  point  on  the 
walk  with  respect  to  the  time.  This  point  is 
conveniently  chosen  as  the  point  on  the  sidewalk 
directly  in  front  of  the  house,  because  we  know  this  distance  at  the  time, 
we  wish  to  determine  D<y.  Let  his  distance  from  this  point  be  x.  Then 
Dtx  =  4.4. 

To  find  the  relation  between  these  two  derivatives  we  must  first  express 
2/  as  a  function  of  x.    From  the  figure  We  have,  at  any  time, 


y  =  V(x2  +  49)  =  (a;2  +  49)§, 
and  hence,  by  Theorem  5,  page  270,  differentiating  with  respect  to  t, 

Dty  =  Ux'  +  49)-ii).(x2  +  49)  =  (^^^  • 

We  are  told  that  when  x  =  24,  DtX  -  4.4,  and  hence, 


Dty 


24  X  4.4        24  X  4.4 


(576 +  49)  J 


25 


=  4.2  feet  per  second. 


EXERCISES 

1.  In  Example  1  above,  find  the  rate  at  which  the  circumference  is 
increasing  when  the  radius  is  6  inches.  Find  also  the  rate  at  which  the 
area  and  circumference  are  increasing  when  the  radius  is  10  inches.  What 
essential  difference  is  there  in  the  rates  at  which  the  circumference  and  area 
increase? 

2.  Imagine  a  belt  stretched  around  the  earth  at  the  equator  (radii 
3963  miles).    If  the  radius  of  the  belt  increases  uniformly  at  the  rate  o! 
3  feet  per  second,  how  much  will  the  circumference  increase  in  the 
second? 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS     291 

3.  A  man  6  feet  tall  walks  directly  away  from  a  lamp  post  12  feet  high 
at  the  rate  of  4  miles  an  hour.  How  fast  does  the  shadow  of  his  head 
move? 

4.  A  man  is  walking  at  the  rate  of  3  miles  an  hour  along  a  sidewalk 
which  is  8  feet  from  the  buildings  beside  it.  A  hght  on  the  other  side  of 
the  street,  30  feet  from  the  walk,  casts  a  shadow  of  the  man  on  the  build- 
ings.   How  fast  does  his  shadow  move? 

5.  A  cruiser  is  moving  parallel  to  a  straight  coast  at  a  distance  of  6 
miles  from  it,  at  the  rate  of  27  miles  an  hour.  How  fast  is  the  cruiser  ap- 
proaching a  fort  on  the  shore,  15  miles  from  the  point  directly  opposite 

'  the  ship? 

6.  The  pressure  and  volume  of  a  gas  at  a  constant  temperature  satisfy 
!  the  relation  pv  =  k.    At  a  certain  time  the  volume  is  3  cubic  feet,  the 

pressure  is  10  pounds  per  square  foot,  and  the  volume  is  increasing  at 
I  the  rate  of  0.2  of  a  cubic  foot  per  second.     Find  the  rate  of  change  of 

the  pressure. 
I       7.  Two  ships  start  from  the  same  place  at  the  same  time.    One  sails 
I  east  at  12  knots  an  hour,  the  other  south  at  18  knots  an  hour.     How  fast 
i  will  they  be  separating  after  half  an  hour? 

I  8.  The  height  of  a  ball  thrown  vertically  upward  with  a  velocity  of 
64  feet  per  second  is  given  by  s  =  64i  -  16^^^  The  rays  of  the  sun  make 
an  angle  of  30°  with  the  horizontal.  How  fast  is  the  shadow  of  the  ball 
moving  along  the  ground  just  before  the  ball  hits  the  ground? 

103.  Small  Errors.  In  measuring  any  quantity  directly, 
we  usually  have  some  idea  of  the  accuracy  of  the  measure- 
ment. Thus  we  may  measure  the  length  of  the  edge  of  a  cubi- 
cal box  and  feel  confident  that  the  length  is  10  inches  with  an 
error  of  not  more  than  J  of  an  inch.  This  is  indicated  by  say- 
ing that  the  length  is  10  =»=  |  inches. 

Frequently  the  size  of  the  error  is  of  less  importance  than 
the  relative  error,  which  is  the  ratio  of  the  error  to  the  quantity 
measured.  Thus  the  relative  error  in  the  edge  of  the  cube 
labove  is  the  ratio  of  |  to  10,  that  is,  ^V  =  0.0125  or  1.25%. 

The  relative  error,  rather  than  the  magnitude  of  the  error 
itself,  indicates  the  degree  of  accuracy  of  a  measurement. 
[Nothing  can  be  said  of  the  comparative  accuracy  of  the  measure- 
ments of  the  lengths  of  two  lines  if  it  is  known  that  the  error 
lin  one  case  is  2  feet  and  that  in  the  other  it  is  only  half  a  foot. 
If  the  first  error  occurred  in  measuring  a  side  of  a  farm  half  a 
naile  long,  the  relative  error  would  be  2^\ji  =  0.000758,  which 


292  ELEMENTARY  FUNCTIONS 

is  less  than  0.08  of  one  per  cent,  while  if  the  second  error  waj 
made  in  measuring  the  frontage  of  a  city  lot  50  feet  wide  th( 
relative  error  would  be  0.5/50  =  0.01,  or  one  per  cent.  Undei 
these  circumstances  the  first  measurement  would  be  by  fai 
the  more  accurate. 

Many  quantities  are  measured  indirectly.  Thus  to  find  th( 
volume  of  a  cubical  box  we  measure  the  length  of  an  edge 
and  compute  the  volume.  If  the  length  of  the  edge  is  fouud  t( 
be  10  inches,  then  the  volume  is  F  =  1000  cubic  inches.  If  an 
error  of  |  of  an  inch  is  made  in  measuring  the  edge,  what 
the  error  in  the  volume?  In  the  example  below  we  shall  de 
velop  a  method  of  answering  such  a  question. 

Example.  The  side  of  a  square  was  found  by  measuring  to  be  3  inches 
If  the  measurement  is  0.1  of  an  inch  too  small,  what  is  the  approximate 
error  in  the  computed  area? 

First  solution.     It  is  obvious  that 

the  computed  area  =  9 
and  the  true  area  =  (3  +  O.l)^  =  9  +  0.6  +  0.01. 

Since  0.01  is  small  in  comparison  with  the  other  terms,  it  may  be  di 
regarded,  and  the  approximate  error  is  0.6. 

Not  only  is  the  term  disregarded  much  smaller  than  the  others,  but  i1 
is  smaller  than  the  change  in  the  approximate  error  due  to  a  slight  change 
in  the  estimated  error  in  the  measurement  of  the  side.  For  if  the  measure- 
ment of  the  side  is  0.11  of  an  inch  too  small,  then 

the  true  area  =  (3  +  0.11)2  ^  9  ^  0.66  +  0.0121, 

and  the  approximate  error  is  now  0.66.  Thus  a  slight  change  in  the 
estimated  error  in  the  original  measurement  of  the  side  produces  a  change 
of  0.06  in  the  original  approximate  error  in  the  area,  and  this  change  is 
6  times  the  term  originally  neglected. 

The  approximate  relative  error  is  the  ratio  of  the  approximate  error  to 
the  computed  area,  i.e.,  0.6/9,  =  0.07  or  7%. 

It  is  instructive  to  first  solve  the  problem  for  any  square  and  then  sub- 
stitute the  given  numbers. 

Let  X  be  the  value  of  the  side  found  by  measurement,  and  Ax  the  esti- 
mated error.  Let  y  denote  the  computed  value  of  the  area,  and  Ay  the 
error  in  y.     Then  the  computed  area  is 

y  ^  x^  ( 

and  the  true  area  is 

y  +  Ay='{x  +  Ax)^  =  x^  +  2xAx  +  Az*  (2) 


• 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    293 


t 

I 

i 
I 

it 


FiQ.  169. 


Subtracting  (1)  from  (2),  the  error  in  the  com- 
puted area  is  seen  to  be 

^y  =  2x^x  +  ^x^.  (3) 

When  Aa;  is  small,  Aa:^  is  very  small  compared  to 
2x^x,  so  that  Ax2  may  be  neglected,  and  the  ap- 
proximate error  in  y  is  2xAa:. 

Equation  (3)  is  readily  interpreted  from  the 
figure.  The  product  xi^x  is  the  area  of  either  of  the 
rectangles,  and  Ax^  is  the  area  of  the  small  square. 

Obviously,  if  Arc  is  small  compared  to  a:,  the  small  square  is  negligible  as 
compared  with  the  two  rectangles.  It  is  also  clear  that  a  shght  change  m 
Ax  produces  a  change  in  the  two  rectangles  which  is  greater  than  the  small 
square  disregarded. 

Denoting  the  approximate  error  by  dy,  we  obtain  from  the  above 
dy  =  2xAx.    The  relative  error  is  therefore 

d^  _  2xLx     ^Ax 


80  that  the  relative  error  in  ?/  is  twice  the  relative  error  in  x. 

Substituting  the  given  values  of  x  and  Ax  we  get,  as  before, 
cZ?/  =  2  X  3  X  0.1  =  0.6,  and  dy/y  =  2  x  0.1/3  =  0.07,  or  7%. 

Second  solution.  The  first  solution  is  useful  for  clarifying  the  ideas  in- 
volved, but  the  method  now  to  be  considered  is  more  convenient  in  all  but 
the  simplest  examples. 

Letting  x  be  the  side  of  any  square,  the  computed  area  is  given 
by  (1),  whose  graph  is  shown  in  Fig.  170.  The  ordinate  y  =>  MP  of 
any  point  represents  the  computed  area  of  a  square  whose  side  as 
measured  is  x  =  OM.  If  the  error  in  the  measurement  is  Aa:  =  MNf 
then  the  ordinate  NQ  represents  the  true  area  and  Ay  =  RQ  the  true 
error  in  y. 

Construct  the  line  tangent  at  P,  and  let  it  cut  the  ordinate  NQ  at  S, 
Let  dy  =  RS.  Then  dy  is  an  approximate  value  of  the  true  error  Ay, 
and  if  Aa:  is  very  small  this  gives  a  very  good  approximation.  For  if 
MN  were  less  than  one-eighth  of  an  inch  in  this  figure,  S  would  be  prac- 
tically coincident  with  Q. 

In  order  to  have  a  more  consistent  notation  set  Ax  =  dx,  so  that  dx  rep- 
resents the  error  in  x. 

Then  in  the  right  triangle  PRS,  we  have  PR  =  dx,  and 


since  tan  0  =  Dxy. 


dy  -  dx  tan  ZRPS 
=  dx  tan  d 
-  D^y  dXf 


(4) 


294 


ELEMENTARY  FUNCTIONS 


From  (1)  we  have  T>xy  =  2x,  and  hence,  from  (4),  dy  =  2x  dx,  the  same 
result  that  was  obtained  in  the  first  solution.     The  error  for  the  given 

square,  and  the  relative  error,  are 
found  as  before. 

If  a  magnitude  x  be  meas- 
ured, and  a  second  magni- 
tude y  be  computed  from  it, 
then  2/  is  a  function  of  x, 
y  =  fix).  The  reasoning  in 
the  second  solution  applies 
equally  well  to  the  graph  of 
this  function,  if  we  regard 
the  curve  in  the  figure  as 
the  graph  of  any  function. 
Hence: 

If  y  =  f(^))  cif^  approxi- 
mate value  of  the  error  in  y 
due  to  an  error  of  dx  in  the 
measurement  of  x  is  the  prod- 
uct of  the  derivative  of  y  with  respect  to  x  and  dx;  i.e., 

dy  =  Dxydx.  (5) 

An  approximate  vahie  of  the  relative  error  is  given  by  dy  /y. 

In  what  follows  we  shall  use  the  terms  error  and  relative 
error  to  denote  these  approximations  unless  the  contrary  is 
explicitly  mentioned.  — ■, 

In  applying  this  method,  it  is  necessary  first  to  compute  y 
as  a  function  of  x  without  using  the  given  numerical  value  of  x. 
For  example,  in  the  illustration  above,  we  must  first  find  the 
area  of  any  square. 

EXERCISES 

1.  The  edge  of  a  cube  is  found  by  measurement  to  be  8.43  inches  with 
a  probable  error  of  0.005  inches.  What  is  the  error  in  the  volume?  The 
relative  error? 

2.  The  side  of  a  square  is  almost  exactly  4  inches.  What  is  the  allow- 
able error  in  the  measurement  of  a  side  if  the  error  in  the  area  is  to  be  de- 
termined to  within  one-tenth  of  one  per  cent? 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS     295 

3.  A  surveyor  measures  a  square  field  with  [a  66-foot  chain  and  finds 
the  area  to  be  40  acres.  If  it  is  found  that  his  chain  has  stretched  and  is 
one  inch  too  long,  find  the  correct  area  approximately. 

4.  Find  the  side  of  a  cubical  box  to  hold  one  quart  (57.75  cubic  inches). 
How  much  variation  may  be  allowed  o«l  a  side  if  the  error  in  the  capacity 
is  not  to  exceed  one  per  cent? 

5.  What  is  the  allowable  error  in  the  measurement  of  the  largest  dimen- 
sion of  a  rectangular  block,  10  x  6  x  4  inches,  if  the  volume  is  to  be  de- 
termined to  within  one-fifth  of  one  per  cent?  What  is  the  allowable  error 
in  the  smallest  dimension?  (Assume  in  each  part  of  the  problem  that  the 
other  two  dimensions  are  exact.) 

6.  The  intensity  of  a  certain  light  at  a  distance  x  from  the  source  is 
/  =  \/x^.     What  is  the  error  in  /  if  a:  =  12,  with  a  probable  error  of  0.1? 

7.  The  quantity  of  heat  necessary  to  raise  the  temperature  of  a  certain 
substance  from  0°  C.  to  6°  C.  is  Q  =  .5290  6  +  .0003  6^.  If  in  an  experi- 
ment B  is  found  to  be  20°,  with  an  estimated  error  of  not  more  than  O'^.S, 
what  is  the  error  in  Qf    The  relative  error? 

8.  The  relation  between  Fahrenheit  and  Centigrade  thermometer  read- 
ings is  i^  =  1.8C  +  32.  A  reading  of  19°.2  is  made  on  a  Centigrade  ther- 
mometer, and  the  temperature  Fahrenheit  computed  from  it.  Find  the 
percentage  of  error  in  the  computed  value,  if  the  error  in  the  reading  is 
OM. 

9.  The  depth  of  the  Colorado  Canyon  at  a  certain  point  was  found  from 
the  relation  s  =  16^^  by  dropping  a  stone  and  observing  the  time  of  falling. 
Is  the  percentage  of  error  in  the  computed  depth  less  than  or  greater  than 
the  percentage  of  error  in  the  measurement  of  the  time? 

10.  The  distance  to  the  sun  as  determined  by  recent  measurements  has 
been  stated  to  be  92,780,000  ±  500,000  miles  (the  notation  indicating  that 
the  error  is  probably  not 
greater  than  500,000  miles). 
One  method  of  computing  this 
distance  is  indicated  in  the 
figure,  where  a  is  the  angle  of 
parallax,  8.811",  at  the  sun 
subtended  by  the  radius  of  the 
earth.  What  is  the  allowable 
percentage  of  error  in  the  radius  to  give  the  above  result  for  the  distance 
to  the  sun? 

The  radius  of  the  earth  is  found  by  measuring  an  arc  of  one  degree,  or 
0.017453  radians,  on  the  equatorial  or  on  a  meridian  circle,  and  applying 
the  theorem  on  page  171.  Assuming  that  the  angular  measurement  is 
exact,  what  is  the  allowable  percentage  of  error  in  the  measurement  of  the 
arc? 

An  arc  of  one  degree  has  been  measured  recently  by  using  a  brass  rod 


Fig.  171. 


296 


ELEMENTARY  FUNCTIONS 


15  feet  long  with  a  probable  error  of  0.00012  inches.    Is  this  suflficiently 
accurate  for  the  computation  of  the  distance  to  the  sun? 

Why  is  it  that  the  distance  to  the  sun  cannot  be  determined  with 
greater  accuracy? 

11.  If  1/  is  a  linear  function  of  x,  y  =  mx  ^  b,  is  the  percentage  of  error 
in  the  computed  value  of  y  ever  less  than  the  percentage  of  error  iu  the 
measured  value  of  x? 

12.  The  radius  and  an  arc  of  a  circle  were  measured,  and  found  to 
be  6  inches  and  15  inches  respectively,  and  the  number  of  radians  in  the 
central  angle  subtended  by  the  arc  was  computed  (Theorem,  page  171). 
Find  the  error  and  the  relative  error  in  the  angle  due  to  an  error  of  one- 
tenth  of  an  inch  in  the  radius;  in  the  arc;  assuming  in  each  case  that  the 
other  measurement  is  exact. 

104.  Approximate  value  of  f(x  +  Ax),  li  y  =  f{x),  it  is 
frequently  convenient  to  use  f(x)  to  denote  the  derivative, 
instead  of  D^y.  The  latter  notation  is  used  when  we  are  de- 
noting a  function  by  y,  the  former  when  we  are  talking  of  a 
function /(a;),  as  in  this  section. 

If  the  graph  of  f(x)  is  drawn,  and  if  x  =  OM  and  Ax  =  M  N, 
then  the  value  of  f{x  +  Ax)  is  represented  by  the  ordinate  NQ, 

In  the  preceding  section  it  was  seen 
that  if  Ax  is  small  dy  =  RS  is  a,  good 
approximation  of  Ay  =  RQ.  Hence 
if  Ax  is  small  NS  is  a  good  approxi- 
mation of  NQ  =  f{x  +  Ax). 

Since  RS  =  Ax  ism  d  =  Ax  f(x) 
and  since  NR  =  MP  =  /(x),  we 
have,  approximately 

f(x  +  Ax)  =  NR  +  RS 

=  fix)  +  fix)  Ax. 

Hence  we  have  the 
Theorem.    If  Ax  is  small,  an  ap- 
proximate value  of  fix  +  Ax)  is  given  by  the  relation  fix  +  Arc) 
=  fix)  +  fix)  Ax,  where  fix)  denotes  the  derivative  of  fix). 

In  more  advanced  work  in  mathematics  the  error  in  this  ap- 
proximation is' considered,  and  also  better  approximations  in- 
volving the  second  and  higher  powers  of  Ax.  The  utility  of 
the  approximation  is  seen  in  the  examples  following. 


Fig.  172. 


i 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS     297 


Example  1.     Find  an  approximate  value  of  3.01'. 

Since  3.01  may  be  put  in  the  form  3  +  0.01,  we  have  3.01  =  x  +  Ax, 
where  x  =  3  and  Ax  =  0.01.  And  if  fix)  =  x',  the  required  number  is  the 
value  of /(x  4-  Ax).     By  the  Theorem  we  have  approximately 

(x  +  Ax)3  =  x^  +  3x2Ax. 
Then  3.0P  =  3'  +  3  x  3^  x  0.01  =  27  +  0.27  -  27.27. 

A.S  the  exact  value  is  3.01'  =  27.270901,  the  approximate  value  obtained 
by  the  Theorem  is  correct  to  four  significant  figures. 

Example  2.  Compute  a  table  of  squares,  to  two  decimal  places,  for 
the  values  x  =  2.01,  2.02,  2.03,  .  .  .  2.10. 

The  square  of  2.01  may  be  computed  from  the  square  of  2,  the  square  of 
2.02  from  that  of  2.01,  etc.,  by  means  of  the  Theorem  which  shows  that, 
approximately,  (x  +  Ax)^  =  x^  +  2xAx. 

The  computation  may  be  systematized  by  arranging  it  in  tabular  form. 
The  last  column  in  the  table  gives  the  squares  of  the  numbers  in  the  first 
column.  The  accuracy  of  the  square  of  the  last  number  in  the  table 
checks  the  accuracy  of  the  entire  computation. 


x+Ax 

^ 

Ax 

x2  +  2xAx 

(x  +  Ax)2 

2.01 

2 

.01 

4           +.04      =4.04 

4.04 

2.02 

2.01 

.01 

4.04      +.0402  =  4.0802 

4.08 

2.03 

2.02 

.01 

4.0802+  .0404  =  4.1206 

4.12 

2.04 

2.03 

.01 

4.1206  +  .0406  =  4.1612 

4.16 

2.05 

2.04 

.01 

4. 1612 +  .0408  =  4. 2020 

4.20 

2.06 

2.05 

.01 

4.2020+  .0410  =  4.2430 

4.24 

2.07 

2.06 

.01 

4.2430+  .0412  =  4.2842 

4.28 

2.08 

2.07 

.01 

4. 2842 +  .0414  =  4. 3256 

4.33 

2.09 

2.08 

.01 

4.3256+  .0416  =  4.3672 

4.37 

2.10 

2.09 

.01 

4.3672+  .0418  =  4.4090 

4.41 

The  computation  is  carried  to  four  decimal  places,  as  these  places  have 
'  an  accumulative  efifect  which  ultimately  changes  the  value  of  the  tabular 
difference  (at  what  point  in  the  table?),  but  very  few  of  the  digits  in  the 
third  and  fourth  decimal  places  are  accurate.  The  approximations  in  the 
last  column  are  made  by  taking  the  nearest  figure  in  the  second  decimal 
place. 

The  approximation  given  in  the  Theorem  is  much  used  in 
the  computation  of  tables,  as  in  Example  2.  It  cannot  be 
used,  however,  to  construct  a  complete  table  without  making 
use  of  the  limit  of  error  of  the  approximation.    Thus  if  the 


298  ELEMENTARY  FUNCTIONS 

table  above  was  continued  for  2.11,  2.12,  etc.,  it  would  be  found 
that  the  value  obtained  for  2.16^  would  be  too  small  by  0.01. 
In  the  very  simple  function  x^  the  accimiulative  effect  of  the 
difference  between  (x  +  Ax)^  and  the  approximation  x^  +  2xAx 
may  be  seen  very  clearly  (see  Exercise  2  below).  But  for 
most  functions  the  question  of  the  error  involved  is  much  more 
complicated. 

EXERCISES 

1.  Compute  an  approximate  value  of  each  of  the  functions  below  for 
the  given  value  of  the  variable. 

(a)  x\  6.2.  (b)  x\  4.1.  (c)  V^  2.1.  (d)    1/x,  2.1. 

2.  Compute  the  squares  of  1.01,  1.02,  and  1.03  using  (a)  the  approxima- 
tion {x  +  AxY  =  x^  +  2x^x;  (b)  the  exact  relation  {x  +  AxY  =  x^  +  2x  ^  + 
Ax^.  Note  the  error  in  the  approximation  of  1.03^  due  to  the  accumulated 
effect  of  neglecting  Ax^  in  the  approximation. 

3.  Compute  a  three-place  table  of 

(a)  Squares  for  x  =  3.01,  3.02,  •  •  •,  3.10. 

(b)  Cubes  for  x  =  4.01,  4.02,  •  •  -,  4.05. 

(c)  Square  roots  for  x  =  0.26,  0.27,  •  •  •,  0.30. 

(d)  Cubes  for  x  =  2.01,  2.02,  •  •  •.  How  far  can  the  computation 
be  carried  before  the  accumulative  effect  of  the  error  in  the  approximation 
becomes  evident? 

MISCELLANEOUS  EXERCISES 

1.  The  chord  of  the  parabola  y  =  ax^  through  the  focus  perpendicular 
to  the  axis  of  symmetry  is  called  the  latus  rectum.  Find  its  length,  and 
show  that  the  tangents  at  its  extremities  are  perpendicular. 

2.  If  Pi  is  any  point  on  y  =  x",  M  its  projection  on  the  y-axis,  and  T 
the  point  of  intersection  of  the  y-axis  and  the  hne  tangent  at  Pi,  show  that 
TM  =  nOM.    Illustrate  by  figiu-es  for  n  >  1,  0  <n  <  1,  n  <0. 

3.  If  the  tangent  and  normal  to  the  parabola  y  =  ax^  at  Pi  cut  the 
y-axis  at  T  and  N  respectively,  then  the  focus  F  is  equidistant  from  Pi, 
T  and  N. 

4.  If  the  tangent  at  Pi  to  the  hyperbola  xy  =  a  cuts  the  x-axis  at  jT, 
show  that  OPi  =  TPi. 

6.  If  the  normal  at  Pi  to  the  hyperbola  xy  =  a  (a>0)  cuts  the  bisector 
of  the  first  and  third  quadrants  at  B,  then  OPi  =  BPi. 

6.  The  line  y  =  mx  +  \a  passes  through  the  focus  (0,  la)  of  the  parabola 
y  =  ax^.  Find  the  abscissas  of  the  points  at  which  it  intersects  the  parab- 
ola. Show  that  the  tangents  at  these  points  are  perpendicular,  and  that 
they  intersect  on  the  directrix. 


DIFFERENTIATION  OF  ALGEBRAIC  FUNCTIONS    299 

7.  If  Vq  is  the  volume  of  a  quantity  of  water  at  0°  Centigrade  then  the 
volume  at  a  temperature  0°  Centigrade  is  given  by 

V  =  Fo(l  -  0.00005758  d  +  0.000007560^  -  0.0000000351  0^). 

Show  that  the  volume  is  least,  and  hence  the  density  greatest,  when 
6  =  3°.92C.  (Let  the  decimals  be  denoted  by  a,  b,  c  respectively  until 
the  end  of  the  computation,  and  write  0.00005758  in  the  form  5.758  X 
10-5,  etc.) 

8.  A  bandit  is  walking  along  a  street  at  the  rate  of  3  miles  an  hour  toward 
the  intersection  with  a  second  street  making  an  angle  of  45°  with  the  first. 
A  tree  stands  100  feet  from  the  intersection  and  25  feet  toward  the  second 
street  from  the  first.  A  timid  citizen  walks  along  the  second  street  en- 
deavoring to  keep  the  tree  between  himself  and  the  bandit.  How  fast 
does  he  walk? 

9.  There  are  two  sources  of  fight  at  the  points  A  and  B.  At  a  distance 
X  from  the  first  the  illumination  is  /i  =  S/x^,  and  at  a  distance  x  from  the 
second  the  illumination  is  /2  =  27/a;2,  Find  the  point  on  the  segment  AB 
where  the  sum  of  the  illuminations  is  a  minimum. 

10.  Suppose  that  Ui,  a-i,  as,  .  .  .  a„,  are  the  values  of  n  measurements 
of  a  magnitude  whose  true  value  is  x.  Then  the  errors  in  the  measure- 
ments are  respectively  x  -  ai,  x  -  a2,  x  —  az,  .  .  .  x  -  an,  some  of  which 
are  positive,  and  some  of  which  are  negative.  The  theory  of  least  squares 
asserts  that  the  most  probable  value  of  a:  is  such  that  the  sum  of  the  squares 
of  the  errors  is  a  minimum.  Show  that  the  most  probable  value  of  x  is 
the  arithmetic  mean  (or  average). 

11.  Find  the  dimensions  of  the  strongest  beam  which  can  be  sawed 
from  a  log  12  inches  in  diameter,  assuming  that  the  strength  of  the  beam 
varies  as  the  breadth  and  the  square  of  the  depth. 

12.  Find  the  dimensions  of  the  stiflfest  beam  which  can  be  cut  from  a 
log  10  inches  in  diameter,  if  the  stiffness  is  proportional  to  the  breadth  and 
the  cube  of  the  depth. 

13.  Plnd  the  dimensions  of  the  rectangle  with  maximum  perimeter 
which  can  be  inscribed  in  a  circle  of  radius  r. 

14.  A  point  moves  along  the  straight  fine  y  =  2x  +  10.  If  its  abscissa 
is  increasing  at  the  rate  of  3  inches  per  second  when  x  =  1,  find  the  rate 
at  which  its  ordinate  changes. 

15.  Two  railroad  tracks  intersect  at  right  angles.  A  train  on  one  track 
is  24  miles  from  the  intersection  and  is  approaching  it  at  the  rate  of  30 
miles  an  hour.  A  train  on  the  other  track  is  7  miles  from  the  intersection, 
and  is  receding  from  it  at  45  miles  an  hour.  Is  the  distance  between  the 
trains  increasing  or  decreasing?     How  rapidly? 

16.  A  16  foot  ladder  resting  against  the  side  of  a  barn  begins  to  sfip. 
When  the  foot  of  the  ladder  is  10  feet  from  the  bam  it  is  moving  2  feet 
per  second.    How  fast  is  the  top  of  the  ladder  moving? 


300  ELEMENTARY  FUNCTIONS 

17.  The  side  of  a  square  is  39.51  inches,  with  ap  error  of  not  more  than 
0.01.  Find  the  error  in  the  area  of  the  square.  Is  the  value  of  39.51^ 
given  in  Huntington's  Tables  suflSciently  accurate  for  the  area? 

18.  The  intensity  of  heat  at  a  distance  of  x  feet  from  a  source  of  heat  is 
/  =  lOO/a:^.  If  a  body  moves  directly  away  from  the  source  at  the  uni- 
form rate  of  3  feet  per  second,  how  rapidly  is  the  intensity  of  the  heat 
changing  when  x  =  h  feet?    When  a;  =  20  feet? 

19.  The  side  of  an  equilateral  triangle  is  5.4  ±0.1  inches.  What  is 
the  percentage  of  error  in  the  area  of  the  triangle? 

20.  If  a  body  moves  so  that  v^  =  k8^  show  that  the  acceleration  is  con- 
stant. An  automobile  moving  v  miles  per  hour  on  a  slippery  pavement 
should  be  able  to  stop  in  s  feet,  where  v*  =  17s.  Find  the  acceleration  in 
feet  per  second  per  second. 

21.  A  rectangular  grain  bin,  without  a  cover,  is  to  be  divided  into  two 
equal  parts  by  a  partition  parallel  to  the  ends.  The  bin  is  to  be  3  feet 
deep  and  to  hold  98  cubic  feet.  Find  its  dimensions  if  the  amount  of 
lumber  required  is  a  minimum. 

22.  A  rectangular  pan  is  to  have  its  width  two-thirds  of  its  length  and 
its  capacity  is  to  be  {^^  of  a  cubic  foot.  Find  its  dimensions  if  the  amount 
of  tin  used  is  a  minimum. 

23.  The  thrust  of  an  aeroplane  is  given  by  the  equation  (see  Exer- 
cise 11,  page  157) 

*  =  TF(0  +  ig). 

Find  the  value  of  B  for  which  ^  is  a  minimum.  Substitute  this  value 
of  Q  in  the  given  equation,  and  determine  how  t  varies  as  the  fineness  / 
changes. 

24.  The  power  required  for  oblique  flight  of  an  aeroplane  up  a  slope 
m  is 

T  =m  (^  +yl^)  +  ^^^  =  ^^(^  +  j4  +  '^)' 

^  For  gliding  flight  T  =  0;  what  relation  must  eonnect  </  and  m  for  gliding 
flight?  \i  Q  =  Q\  makes  m  a  minimum,  mi,  show  that  m\  =  20i.  What 
relation  exists  between  m  and  the  horizontal  and  vertical  distances  the 
aeroplane  glides?  How  does  the  fineness  /  affect  the  greatest  horizontal 
distance  the  aeroplane  can  glide?  Show  that  the  aeroplane  can  glide  along 
a  given  slope  for  two  different  values  of  the  angle  of  incidence  Q  and  hence 
for  two  different  velocities  (see  Note,  page  156). 


CHAPTER  VII 
INTEGRATION 

105.  Introduction.  In  Chapter  VI  we  considered  the 
problem :  Given  the  function  y,  to  find  the  derivative  D^y. 

In  this  Chapter  we  shall  consider  the  inverse  problem :  Given 
the  derivative  DxVj  to  find  the  function  y. 

Definition.  A  function  whose  derivative  is  a  given  func- 
tion is  called  an  integral  of  the  given  function  and  the  process 
of  finding  it  is  called  integration. 

Thus,  if  Z>x/(x)  =  F(x),  then  }{x)  is  an  integral  of  F{x). 

Example  1.     Given  Dxy  =  2a:,  find  y. 

Since  DxX"  =  wic""^,  we  have  DxX^  =  2x.  Hence  x^  is  an  integral  of  2x. 
But  the  derivative  of  each  of  the  functions  x"^  +  5,  x^  -  7,  x^  +  f ,  is  2x. 
Hence  these  functions  are  also  integrals  of  2x.  In  fact,  any  function  of 
the  form  y  =  x^  +  C,  where  C  is  any  constant,  has  the  derivative  2a:,  and 
is  therefore  an  integral  of  2x. 

Thus  we  see  that  the  process  of  differentiating  the  function  x"^  gives 
rise  to  the  single  derivative  2x,  but  that  the  inverse  process  of  integrating 
2x  gives  rise  to  an  indefinite  number  of  integrals  x"^  +  C  which  differ  by  a 
constant. 

Theorem.  If  two  functions  f(x)  and  F{x)  have  the  same  derivor 
tive^  their  difference  is  a  constant. 

Let  y  =  fix)  -  F{x). 

Then  D,y  =  DJ{x)  -  DJ'ix), 

By  hypothesis,  DJ{x)  =  D^Fix). 
Hence  Dxy  =  0. 

But  a  function  whose  derivative  is  equal  to  zero  for  all  values 
of  the  variable  is  a  constant.  For  the  tangent  line  at  e very- 
point  on  the  graph  is  horizontal,  so  that  the  graph  of  the  func- 
tion is  a  straight  line  parallel  to  the  a;-axis. 

301 


302 


ELEMENTARY  FUNCTIONS 


Hence  y  =  f{x)  -  F{x)  =  C, 

or  fix)  =  Fix)  +  C. 

That  is,  every  integral  can  be  obtained  from  a  given  integral 
by  adding  the  proper  constant. 

The  constant  C  which  is  added  to  a  known  integral  of  a  func- 
tion is  called  the  constant  of  integration  and  it  should  not  be 
omitted  in  performing  the  operation  of  integration. 

If  y  is  a  function  of  x  whose  derivative  with  respect  to  a;  is 
given,  say  Fix),  then 

D.y  =  Fix). 

The  notation  employed  to  indicate  that  2/  is  an  integral  of 
Fix)  is 

y  =  fFix)dx, 

the  equation  being  read  ''y  equals  the  integral  of  Fix)dxy    In 
the  illustration  above,  since  Dzy  =  2x,  then 

y  =  f2x  dx  =  x'^  +  C, 

In  a  particular  problem  sufficient  data  are  usually  given  to 
determine  the  constant  of  integration  and  to  enable  us  to  find 
a  particular  integral  which  satisfies  the 
given  conditions.  The  graphical  signi- 
ficance of  the  constant  of  integration 
and  the  method  of  determining  its  nu- 
merical value  so  as  to  satisfy  given 
conditions  is  shown  in 

Example  2.  Find  y  if  DxV  =  z'^  -  4  and 
interpret  the  result  graphically.  Find  the 
equation  of  that  one  of  the  resulting  curves 
which  passes  through  the  point  (1,  2). 

By  dififerentiating  a^  we  obtain  3x^,  and 
hence  we  must  differentiate  Ix^  in  order  to 
get  x^.    Differentiating  4x  gives  4,  and  hence  if 

D,y  =  x^-4, 
y='ix^-4x  +  C, 

where  C  is  the  constant  of  integration. 

This  represents  a  set  of  curves  which  may  be  obtained  from  one  of  them  I 
by  moving  it  up  or  down  (Theorem,  page  19) .  Several  curves  of  this  set] 
are  shown  in  the  figure. 


1          y           1/ 

-i     %       7^ 

/  />     ^            // 

4       ^    ^/X 

t-^^-'\       LL 

'A-n  \«../, 

3    ^  ^2n* 

i/f?^  °\  '  FT 

A~\\A~ 

i    'A    t 

^  J- 

::----!£ 

Fia.  173. 


INTEGRATION  303 

These  curves  have  parallel  tangent  lines  at  points  with  the  same  ab- 
scissas, since  the  derivative  of  y  with  respect  to  x  is  the  same  for  each  fimc- 
tion,  namely,  Dxy  =  x^  —  4. 

The  intercepts  of  these  curves  on  the  2/-axis  are  the  respective  values  of  C. 
If  (1,  2)  lies  on  one  of  these  curves  then  2  =  i-l'-4-l  +  C,  whence  C  =  5| 
and  the  required  equation  is  i/  =  i^^  -  4a:  +  5|. 

Example  3.    If  Dxy  =  ax**,  show  that  y  =         ..x"'''^  +  C. 

The  result  is  correct  because  if  we  differentiate  it  we  get 

D^y  =  ^j^T^i  (n  +  l)x''  =  ox". 

Hence,  to  integrate  a  term  of  the  form  ax^'j  increase  the  exponent 
by  one  and  divide  the  coefficient  by  the  new  exponent. 

In  like  manner,  if  Dxy  =  aw"  D^u,  then  y  = 1^"+^  +  C. 

n  -f-  JL 

If  the  integral  notation  is  used,  these  results  may  be  written 
Tax"  dx  =  — ^  a:"+i  +  C,         faW"  D^u  dx  =  — ^  w"+i  +  C. 

Example  4.     (a)  Find  y  if  Dxy  =  4:3^. 

4 
From  the  preceding  example  we  have  y  =  ^ — z  x*  +  C  <=  x*  +  C. 

(b)  Integrate  f(Sx^  +  Sx)dx. 

We  have  f{Sx^  +  Sx)dx  =  fSx^  dx  +  fSx  dx  =  x^  -\- 4x^ -{-  C. 

(c)  Find  y,  if  Dxy  =  i{x^  +  Sx)i  (2x  +  3). 
If  we  think  of  x^  +  3a:  as  a  function  u,  then  2a:  +  3  is  DxU,  and  therefore 

y  =  l^-^i^-^c={x^  +  3x)i  +  c. 

Differentiation  is  a  direct  process  while  integration  is  an  in- 
verse process.  The  former  can  be  carried  out  according  to  a 
general  method  of  procedure  biit  there  is  no  general  method  of 
procedure  in  the  case  of  the  latter.  Integration  in  the  last 
analysis  is  a  matter  of  trial  in  which  attempts  are  made  to  re- 
duce the  given  function  to  a  form  the  integral  of  which  is  known. 
A  thorough  knowledge  of  differentiation  is  essential  for  the 
process  of  integration. 

EXERCISES 

In  Exercises  1-12,  find  y  if  Dxy  is  the  given  function,  and  illustrate 
the  result  graphically.  Determine  the  equation  of  that  member  of  the 
set  of  curves  which  passes  through  the  indicated  point. 


304 


ELEMENTARY  FUNCTIONS 


1. 

4. 

x  +  1,  (1,  2). 
x3  +  a:,  (- 1,  11). 

2.  x2  -  3,  (-  1,  3). 
5.  A  (3,  4). 

3.  x^-a:,  (0,2). 
6.   ^7^,  (1,  3). 

7. 

Va^  +  1,  (4,  -2). 

8.  ^„  (1,  1). 

9.    ^-,  (2,  1). 

10 

^      ,  (0,  ^). 

11.   — ^,  (2,  4). 

y/x^  -  4 

12.        2x-3 

V2a:  +  3'  '       ' 

V(a:2  _  3^)3'  U>  "^ 

13. 
14. 

For  what  value  of  n  does  the  result  in  Example  3  above  fail? 
Given  D:,y  =  (x^  +  3x-  2Y  {2x  +  3),  find  y. 

I  

.    15.  Given  D^y  =  {Sx^  +  4Wix^  +  ^)\  find  y. 

16.  Find  the  equation  of  the  curve  whose  slope  at  any  point  is  2a;  +  3 
and  which  passes  through  the  point  (2,  3).  Find  the  equation  of  the  tan- 
gent hne  at  this  point.     Plot  the  curve. 

17.  The  slope  of  a  curve  at  any  point  is  6x^  +  \x  and  it  passes  through 
the  origin.  Find  the  equation  of  the  curve  and  the  equation  of  the  nor- 
mal line  at  the  point  of  inflection.     Plot  the  graphs  of  the  two  equations. 

18.  Find  the  cubic  function  whose  graph  has  a  maximum  point  at 
(1,  4)  and  a  minimum  point  at  (3,  -  5).  Find  the  equation  of  the  tangent 
line  at  the  point  of  inflection.     Suggestion.     D^y  =  a{x  -  l)(x  -  3). 

19.  The  slope  of  a  curve  at  any  point  is  4a:  -  3  and  it  passes  through 
the  point  (1,  2).    Find  the  equation  of  the  curve  and  the  coordinates  of 

the  minimum  point.    Plot  the 
graph. 

20.  Integrate  the  following: 

(a)  y(6x2  +  8x  -  h)dx. 

(b)  fxl  dx. 

(c)  f^y/x  -  2  dx. 

(d)  r|  dx. 


V 

I 

1 

fi 

Q 

/ 

Ay 

s 

J 

/^ 

A 

AA 

M 

N 

R 

"d 

< — a > 

r 

r<Aa>i 

1          ' 

(e)  f2V4:-3xdx. 

106.  Area  under  a 
Curve.  Consider  the  an 
A,  bounded  by  any  curvl 
y  =  fix)  7  the  X-axis,  the 
fixed  ordinate  MB  at 
X  =  a,  and  the  movinji 
ordinate  at  NP. 
Since  A  changes  as  x  changes  and  is  determined  when  x  h 
fixed,  A  is  a  function  of  x. 

As  N  moves  to  R,  x  increases  by  an  amount  Ax,  A  by  ai 
amount  PNRQ  =  AA,  and  y  by  an  amount  SQ  =  Ay. 


Fig.  174. 


INTEGRATION 


305 


Since  PNRQ  is  less  than  the  rectangle  TNRQ  and  greater 
than  the  rectangle  PNRS,  we  have 

PNRS<AA<TNRQ 

or 

yAx<AA  <(y  +  Ay)  Ax. 

Dividing  by  Ax,  we  have 

AA 

y<-^<y  +  ^y' 

As  Ax  approaches  zero,  y  +  Ay  approaches  y  and  A^  /Ax  ap- 

AA 
proachesDxA;  and  as -^  lies  between  the  magnitudes  y  and 

y-\-Aywe  have  DxA  =  y  =  fix). 

Therefore  we  have  the 

Theorem.     The  rate  at  which  the  area  A  changes  with  rested 
to  X  is  eqaal  to  the  rightrhand  ordinate  o}  the  hounding  curve. 

Symbolically  J  D^A  =  y.  (1) 

The  area  A  can  now  be  found  as  a  function  of  x  by  integra- 
tion, the  constant  of  integration  being  determined  by  the  fact 
that  A  =  0  when  x  -  a.  If  a  fixed 
right-hand  boundary  is  chosen  the 
area  is  determined.  The  method  of 
finding  the  area  is  shown  in 

Example  1.  Find  the  area  bounded  by 
the  curve  y  =  x^,  the  x-axis  and  the  ordi- 
nates  at  x  =  1  and  x  =  4. 

By  the  theorem,     Z>xA  =  x^.  (2) 

.      x' 


Integrating, 


+  C. 


(3) 


Since  A  =  0  when  x  =  1,  we  have  0  =  ^  +  C 
or  C  =  -  i 

.     ^     1 


^               X 

u             Si 

-U 

18                                      M 

M 

15           t^ii 

u 

12                                 ^'- 

9-                           t 

J 

^                      J- 

z 

3                    A 

^s            -W^ 

-  ^  ^'^      2        i       ~V          X 

■""^i-                : 

Hence 


(4) 


Fig.  175. 


Equation  (4)  gives  the  area  under  the  curve  starting  at  the  ordinate  MP, 
at  X  =  1,  and  continuing  to  any  second  ordinate.  In  this  case  the  second 
ordinate  is  fixed  at  x  =  4.     Hence  substituting  x  =  4  in  (4)  we  have 


.43      1 
"*-  3~3 


55  =  21 
3      ^^' 


(5) 


306 


ELEMENTARY  FUNCTIONS 


M         N 
Fig.  176. 


The  graph  of  (4)  is  the  curve  LMD  which  crosses  the  a:-axis  at  x  -  1 
where  the  area  MPBN  begins. 

The  number  of  square  units  in  MBPN  is  the  same  as  the  number  of 
linear  units  in  the  ordinate  ND,  which  is  21  units  in  length. 

Equation  (1)  may  be  interpreted  thus:  Suppose  that  we  had 
y  =  c,  whose  graph  is  a  straight  hne  parallel  to  the  a;-axis. 

Then  the  rate  of  change  of  A  with 

respect  to  x,  DxA,  would  be  uniform, 

since   DJi  =  y  =  c.      It    would    be 

B         D  measured  by  the  change  in  A  due  to 

a  unit  change  in  x  (page  48),  that  is, 

by  a  rectangle  with  base  unity  and 

altitude  c. 

"?J'         Returning  to  the  figure  used  in 

proving  the  theorem,  let  NR  =  1. 

Then    the    area    of    the    rectangle 

PNRS  =  y  X  I  =  y  is  the  amount  by  which  A  would  increase 

when  X  increases  by  unity,  provided  that  A  increased  uniformly 

for  x>ON. 

The  method  of  proving  the  theorem  above  may  be  used  to 
show  that  an  area  A  bounded  by  a  curve,  the  2/-axis,  and  two 
abscissas  (one  fixed,  the  other  not),  is  such  that 

DyA  =  X.  (6) 


EXERCISES 

1.  Find  the  area  under  each  of  the  following  curves  from  x  =  1  to  a;  =  3. 
(a)  y  =  3x2,  (b)  y  =  2/x\         (c)  y  ^  x%,  (d)  ?/  =  x^  +  x  +  7. 

2.  Find  the  area  bounded  by  the  curve  y  =  x^,  the  x-axis  and  the  line 
X  =  2,  and  the  area  bounded  by  the  curve,  the  2/-axis  and  the  line  y  =  4. 
Check  by  finding  the  area  of  the  rectangle  formed  by  the  axes  and  the 
lines  X  =  2  and  y  =  4. 

3.  Find  the  area  above  the  x-axis  and  below  the  parabola  y  =  -  x^  +  4. 

4.  If  the  positive  intercepts  of  the  parabola  y  =-  -  x^  +  2x  +  3  on  the 
coordinate  axes  are  denoted  by  A  and  B,  and  if  a  tangent  parallel  to  the 
chord  AB  cuts  the  ordinates  at  A  and  B  in  C  and  D,  and  touches  the 
parabola  in  E,  show  that  the  area  of  the  parabolic  segment  ABE  is  two- 
thirds  that  of  the  parallelogram  ABCD. 

6.  If  the  tangent  to  the  parabola  ?/  =  3x*  -  6x  +  8,  which  is  parallel 
to  the  chord  joining  the  minimum  point  A  to  the  point  B  on  the  curve 


INTEGRATION  307 

whose  abscissa  is  2,  cuts  the  ordinates  of  A  and  B  m  C  and  D,  find  the 
relation  between  the  area  of  the  parabolic  segment  from  A  \jo  B  and  the 
area  of  the  parallelogram  ABCD. 

6.  Find  the  area  bounded  by  the  following  pairs  of  curves: 

7         15  1 

(a)  y  =  x\  y"  =  x,      (b)  y  =  xi,  y  ^  x\,      (c)  y  =  -  ^x^  +  —  x,  y  =  -• 

7.  Find  the  area  bounded  by  the  curve  y  =  a?  -  %z^  +  9a;  +  4,  the 
tangent  to  the  curve  at  the  point  of  inflection,  and  the  ordinates  at  the 
maximum  and  minimum  points. 

8.  Show  that  the  area  under  the  parabola  y  =  ax^  +  6a;  +  c  between 

x~h  and  x  =  1c,  ik>h),  is  A  =  — = —    (?/i  +  4^2  +  2/3),  where   yi,    y^,  ys 

h  +  k 
are  the  values  of  2/  at  a;  =  A,  a;  =  — 2~>  ^  =  ^  respectively.    Suggestion:  Sub- 
stitute the  values  of  y  in  the  above  expression  and  show  that  the  result  is 
the  same  as  the  area  under  the  curve. 

9.  Find  the  area  under  the  following  parabolas  and  check  the  results 
by  means  of  the  formula  for  the  area  given  in  Exercise  8. 

(a)  y  =  2x^  +  4:X  +  1,    from    a;  =  1  to  a;  =  3. 

(b)  y  =  -  a;2  +  9,     from    a;  =  0  to  a;  =  3. 

(c)  2/  =  3a:2  -  4a;  +  3^     from    x  =  0  to  a;  -  2. 

10.  Show  that  the  area  formula  in  Exercise  8  is  true  for  any  cubic 
function  y  =  ax^  +  bx^  +  cx  +  d. 

11.  Find  the  area  under  the  curve  y  =  a^  +  12a;  +  4  from  a;  =  1  to  a;  =  3. 
Check  the  result  by  means  of  the  formula  in  Exercise  8. 

Definition.  The  average  ordinate  of  a  curve  y  =  f(x)  from 
X  =  a  to  X  =  h  is  the  height  of  the  rectangle  with  base  b  -  a, 
whose  area  is  equal  to  the  area  under  the  curve  from  x  =  a 
to  X  =  b. 

If  y  represents  the  average  ordinate  and  A  the  area  under 
the  curve  from  x  =  ato  x  =  b,  then 

A 


y 


b-a 


12.  Find  the  average  ordinate  of  the  following  curves  for  the  ranges 
indicated. 

(a)  2/  =»  2a;  +  3,  from  a;  =  1  to  a;  =  4. 

(b)  y  =  -x^  +  16,  from  a;  =  0  to  a;  =  4. 

(c)  y  =  1/x^,  from  a;  =  1  to  a;  =  3. 

(d)  y  =  xl,  from  a;  =  0  to  a;  =  4. 


308  ELEMENTARY  FUNCTIONS 

13.  Given  6  =  t^  -  9t^  +  15t  +  30,  where  6  is  the  temperature  at  any 
time  t,  find  the  average  temperature  for  the  range  from  maximum  to  mini- 
mum temperature. 

14.  The  pressure  p  and  the  volume  v  oi  &  gas  are  connected  by  the  equa- 
tion pv^'^  =  k.  and  p  =  15  pounds  per  square  inch  when  v  =  0.5  of  a  cubic  foot. 
Find  the  average  pressure  of  the  gas  in  expanding  from  1  to  3  cubic  feet. 

15.  The  tension  T  oi  a  spring  is  connected  with  the  amount  of  stretch- 
ing s  by  the  equation  T  =  |s  +  3.  Find  the  average  tension  as  s  changes 
from  0  to  3. 

16.  Find  the  percentage  of  error  in  the  area  under  the  curve 
y  =  x^  -2x  +  3  from  x  =  0  to  a:  =  2,  due  to  an  error  of  1  per  cent  in  the 
range. 

107.  Motion  in  a  Straight  Line.  The  fundamental  notions  in- 
volved in  the  motion  of  a  particle  in  a  straight  line  are: 

The  distance,  s,  measured  from  a  convenient  station  on  the 
line. 

The  time,  t,  which  has  elapsed  from  a  fixed  time. 

The  velocity,  v^  given  by  the  equation  v  =  DtS. 
i    The  acceleration,  a,  given  by  the  equation  a  =  DtV. 

The  motion  may  be  described  by  an  equation  involving  two 
or  more  of  the  magnitudes  t,  s,  v,  a. 

We  have  already  considered  motions  described  by  an  equa- 
tion of  the  form  s  =f(t),  and  obtained  the  velocity  and  ac- 
celeration by  differentiation.  We  shall  now  consider  two  other 
types  in  which  integration  is  involved. 

If  the  velocity  is  given  as  a  function  of  the  time  by  an  equa- 
tion of  the  form  v  =  f{t),  the  acceleration,  a,  is  found  by  dif- 
ferentiating while  the  space,  s,  is  found  by  integrating  this 
equation.  The  constant  involved  in  the  integration  is  de- 
termined as  in  the  following  example. 

Example  1.  A  body  moves  from  rest  in  a  straight  line  so  that  its 
velocity  after  t  seconds  is  given  by  the  law  v  =  4<.  Assuming  that  a  ball 
placed  on  a  plane  inchned  at  arc  sin  I  would  roll  according  to  this  law, 
determine: 

(a)  How  far  it  will  roll  in  2  seconds. 

(b)  Its  distance  from  the  upper  end  of  the  plane  after  1.5  seconds,  if  it 
is  placed  4  feet  from  that  end. 

(c)  Its  distance  from  the  lower  end  after  1  second,  if  it  is  placed  12  feet 
from  the  lower  end. 


INTEGRATION 


309 


We  know  that  if  s  represents  distance  from  a  certain  point  at  the  time  t, 
then  Dts  =  V,  and  hence,  in  this  example, 

Dis  =  At. 
The  required  function,  s,  is  therefore  one  whose  derivative  with  respect 
to  t  is  4:t.    One  such  function  is  2t^,  and  adding  the  constant  of  integration 
(page  302),  we  have 

s  =  2f  +  C.  (1) 

The  Value  of  the  constant  of  integration  may  be  determined  if  we  know 
a  pair  of  values  of  t  and  s,  and  these  depend  on  when  and  where  the  ball 
starts  to  roll.  We  shall  assume  that  t  =  0  when  the  ball  begins  to  roll. 
The  distance  s  is  measured  from  some  chosen  station  0. 

(a)  The  distance  the  ball  will  roll  in  2  seconds  is  the  same  as  its  distance 
from  the  starting  point  after  2  seconds.  Hence  for  this  part  of  the  problem 
we  choose  the  station  0  at  the  starting  point,  so  that  s  =  0  when  ^  =  0. 
Substituting  these  values  in  (1)  we  find  that  C  =  0.  Hence,  substituting 
this  value  of  C  in  (1),  the  distance  from  the  starting  point  after  t  seconds  is 

s  =  2P.  (2) 

Then  at  the  end  of  2  seconds,  s  =  2  x  2^  =  8  feet. 

(b)  Take  0  at  the  upper  end  of  the  plane.  Since  the  baU  starts  4  feet 
from  0,  we  have  s  =  4  when  t  =  0.  Substituting  these  values  in  (1),  we 
find  that  C  =  4.  Setting  this  value  of  C  in  (1),  the  distance  from  the  upper 
end  of  the  plane  to  the  ball  after  t  seconds  is 

s  =  2t^  +  4.  (3) 

After  1.5  seconds,  s  =  2(1.5)2  +  4  =  8.5  feet. 

That  is,  the  ball  is  8.5  feet  from  the  upper  end  of  the  plane  after 
1.5  seconds. 

(c)  If  0  is  taken  at  the  lower  end  of  the  plane 
and  the  ball  placed  12  feet  above  it,  then  s  =  -  12 
when  1^=0(8  being  negative  since  the  positive 
direction  for  s  is  the  same  as  that  of  v,  namely, 
down  the  plane).  Substituting  in  (1)  we  find 
C  =  -  12,  so  that  the  distance  from  the  lower  end 
of  the  plane  at  any  time  is 

s  =  2«2  _  12.  (4) 

If  t  =  1,  s  -  -  10,  that  is,  after  one  second  the 
ball  is  10  feet  above  the  lower  end  of  the  plane. 

Graphically,  the  given  relation  v  =  At  is  repre- 
sented by  the  straight  Une  through  the  origin  whose 
slope  is  4.  The  graph  of  (1)  is  given  for  C  =  0, 4,  -  12, 
which  are  the  values  of  C  in  (2),  (3),  (4),  respectively.  The  intercepts  of 
these  curves  on  the  s-axis  are  the  respective  values  of  'C,  which  give  the 
■  distances  from  the  station  0  to  the  ball  when  t  =  0- 


V     , 

i 

\_ 

c- 

r 

l\ 

10 

L 

\\ 

'/ 

\v 

1 

/ 

\\ 

/4 

-T-,  -19 

\  \ 

S' 

'// 

_J 

\ 

0 

^- 

-'\-f 

(y 

:    >', 

Y 

— g 

i 

r~ 

K 

/ 

V    . 

-. 

Fig.  177. 


310  ELEMENTARY  FUNCTIONS 

If  the  acceleration  is  given  as  a  function  of  the  time,  t,  by 
an  equation  of  the  form  a  =  f{t),  the  velocity  is  obtained  by 
integrating  this  equation  and  the  distance  by  integrating  the 
result.  Two  constants  of  integration  are  introduced  in  the 
process  which  are  determined  by  given  conditions  as  in 

Example  2.  A  balloon  is  ascending  with  a  uniform  velocity  of  28 
feet  per  second  and  at  the  height  of  720  feet  a  ball  is  dropped.  When  will 
the  ball  strike  the  ground  and  with  what  velocity? 

Let  s  be  the  height  of  the  ball  above  the  ground,  and  let  ^  =  0  when  the 
ball  is  dropped. 

The  direction  of  the  acceleration  of  gravity,  being  toward  the  center  of 
the  earth,  is  opposite  to  the  positive  direction  of  s,  and  hence  the  accelera- 
tion is  negative. 

The  initial  conditions  and  required  data  are  collected  in  the  table. 
As  soon  as  the  ball  leaves  the  balloon  it  is  subject  to  the  law 

a  =  -  32.  (1) 

Integrating  (1),  »  =  -  S2t  +  d.  (2) 
When  t  =  0,v  =  2S,  and  therefore  Ci  =  28. 

Hence  v=  -  S2t  +  28.  (3) 

Integrating  (3),  s  =  -  16^2  ^  2St  +  d.  (4) 

When  t  =  0,s  =  720,  hence'^Ca  =  720.     Therefore 

s  =  -  16^2  +  2St  +  720.  (5) 

When  the  ball  strikes  the  ground  s  =  0.    Substituting  this  value  of  s  in  (5) 

0  =  -  16^2  ^  2St  +  720, 
or  4<2  -7t-  180  =  0. 

Solving  for  t  we  have 

^     7  =*=  V49  +  2880       54.2      __  , 

t  = 5 =  — ^—  =  6.8  seconds, 

o  o 

the  negative  value  of  t  having  no  meaning  in  this  problem.    Substituting 
this  value  of  t  in  (3) 

«  =  -  32(6. 8)  +  28    -  -  189. 6  feet  per  second. 


EXERCISES 

1.  A  ball  is  rolled  up  a  plane  inclined  at  an  angle  of  10°  with  an  initial 
velocity  of  15  feet  per  second.  Find  the  distance  it  rolls  up  the  plane 
and  the  time  that  elapses  before  it  returns  to  the  starting  point.  (Sug- 
gestion: Find  the 'component  of  the  acceleration  of  gravity  acting  along 
the  plane.) 


t 

s 

V 

a 

0 

720 

28 

-32 

? 

0 

? 

-32 

I 


INTEGRATION  311 

2.  Solve  Example  2  of  the  preceding  section  if  the  balloon  is  descending. 

3.  A  high  jumper  raises  his  center  of  gravity  3  feet.  How  long  is  he  off 
the  ground,  and  with  what  velocity  does  he  hght? 

4.  A  baseball  dropped  from  one  of  the  windows  of  the  Washington 
monument,  500  feet  from  the  ground,  has  been  caught.  Compare  the 
velocity  with  which  it  struck  the  catcher's  hand  with  the  velocity  of  120 
feet  per  second  which  is  said  to  be  the  maximum  velocity  that  a  pitcher 
has  imparted  to  a  ball. 

5.  A  man  descending  in  an  elevator  whose  velocity  is  10  feet  per  second 
drops  a  ball  from  a  height  of  6  feet  above  the  floor.  How  far  will  the 
elevator  descend  before  the  ball  strikes  the  floor  of  the  elevator? 

6.  In  the  preceding  problem,  suppose  the  elevator  is  ascending  instead 
of  descending. 

7.  A  balloon  is  ascending  with  a  velocity  of  24  feet  per  second  when  a 
ball  is  dropped  from  it.  The  ball  reaches  the  ground  in  5  seconds.  Find 
the  height  of  the  balloon  when  the  ball  was  dropped.  Determine  the 
highest  point  that  the  ball  reached. 

8.  An  automobile  reduces  its  speed  from  35  miles  an  hour  to  20  miles 
an  hour  in  8  seconds.  If  the  retardation  is  uniform  how  much  longer  will 
it  be  before  it  will  come  to  rest,  and  how  far  will  it  travel  in  this  length 
of  time? 

9.  How  high  will  a  ball  rise  if  thrown  vertically  upward  with  an  initial 
velocity  of  60  feet  per  second? 

10.  A  street  car  in  going  from  one  stop  to  another  400  feet  distant  is 
uniformly  accelerated  at  the  rate  of  2  feet  per  second  per  second  for  a  dis- 
tance of  320  feet  and  then  brought  to  rest  with  a  uniform  retardation. 
Find  the  time  it  took  to  go  the  400  feet. 

108.  Motion  in  a  Plane.  If  a  particle  moves  along  a  curve 
in  a  plane,  two  rectangular  axes  are  chosen  and  the  position  of 
the  particle  at  any  time  determined  by  means  of  the  coordi- 
nates of  the  particle. 

If  the  coordinates  of  the  particle  on  the  axes  are  given  by 
the  equations 

X  =  m  and  y  =  F{t)  (1) 

the  position  of  the  particle  in  the  plane  is  determined. 

The  discussion  of  the  motion  of  a  particle  in  a  plane  is  thus 
resolved  into  the  discussion  of  two  motions  along  straight  Unes. 

The  components  of  the  velocity,  obtained  by  differentiating 
equations  (1)  are 

Vx  =  DtX    and    Vy  =  Dty.  (2) 


312 


ELEMENTARY  FUNCTIONS 


They  are  represented  in  Fig.  178  by  directed  lines  parallel  to 
the  axes. 

Since  the  components  are  perpendicular  the  magnitude  of 
the  resultant  velocity,  v,  is  given  by  the  equation 


V  =  Vy^x  +  v\ 


(3) 


rit 


'1l 

p 

^ 

/ 

ax       ' 

0 

X 

Fig.  178. 


Fig.  179. 


and  the  direction  of  the  velocity  can  be  found  from  the  equation 

tan  6  =  Vyjvx.  (4) 

The  components  of  the  acceleration,  obtained  by  differentiat- 
ing equations  (2)  are 

ttx  =  DtVx    atid    ay  =  DtVy.  (5) 

They  are  represented  in  Fig.  179  by  directed  lines  parallel  to 
the  axes. 

The  magnitude  of  the  resultant  acceleration,  a,  is  given  by 
the  equation 

a  =  Va^  +  a'y  (6) 

and  its  direction  can  be  found  by  means  of  the  equation 

tan  0  =  Qy  /a^.  (8)| 

Example  1.  If  a  particle  moves  in  accordance  with  the  law  x  =  <', 
y  =  t^,  find  the  equation  of  the  path,  the  position  of  the  particle  when] 
t  =  2,  and  the  magnitude  and  direction  of  v  and  of  a  at  this  point. 

From  the  first  equation  t  =  a;i  and  hence  y  =  (xi)^  =  xi,  which  is  the] 
equation  of  the  path  shown  in  Fig.  180.  Differentiating  the  given  eqiu^l 
tions  with  respect  to  t  we  obtain 


3t\      and 


INTEGRATION 

2t.      (1) 


313 


Hence 


»  =  V9J*  +  4^2  =  < V9i2  +  4    (2) 
and 

tan  0  =  2t/3t^  =  2/3<.       (3) 

Differentiating   equations   (1)   we 
obtain, 

ax  -  Qt,    and    Cy  =  2 
Hence  a  =  VSQt^  +  4, 

and  tan  0  =  2/6t  =  l/3«. 


V 

n 

F 

( 

p.4) 

^ 

r 

— 

-3 

^ 

'^^ 

p] 

a 

^^ 

^ 

N 

^^ 

> 

^/ 

- 

^ 

_ 

u 

^ 

u 

' 

1 

t  VH 

Fig.  180. 


(4) 
'(5) 

(6) 


\ 


Substituting  <  =  2    in    equations   (2),    (3),    (5),    (6),   we  find  v  =  12.64, 
e  =  18°.43,  a  =  12.17,  0  =  9^46,  at  the  point  where  a;  =  8  and  y  =  4. 

The  inverse  problem  of  determining  the  path  of  a  particle, 
given  the  equations  of  the  component  accelerations,  is  illus- 
trated in 

Example  2.  Find  the  equation  of  the  path  of  a  projectile  fired  at  an 
angle  of  30''  to  the  horizontal  with  a  muzzle  velocity  of  1200  feet  per  second. 

Find  the  range  of  the  pro- 
y\  jectile    and    the    maximum 

height  attained. 

Let  the  origin  be  taken  at 
the  initial  point  of  the  path, 
the  a;-and  y-axes  being  hori- 
zontal and  vertical. 

The  horizontal  and  vertical 
components  of  the  initial  velocity  are  1200  cos  30°,  and  1200  sin  30°, 
respectively. 

If  the  resistance  of  the  air  be  disregarded  there  will  be  no  horizontal 
acceleration. 

From  the  instant  the  projectile  leaves  the  gun  gravity  is  acting  on  it 
vertically  in  the  direction  opposite  to  the  positive  direction  on  the  2/-axis. 
Hence  the  acceleration  equations  of  the  motion  are 

a*  =  0  and  ay     =  -  32.  (1) 

Integrating  these  equations  we  have 

Vx  =  Ci    and  Vy  =  -  32t  +  d.  (2) 

When  <  =  0,  tJx  =  1200  cos  30°,  and  Vy  =  1200  sin  30°. 
Therefore  Ci  =  1200  cos  30°,  and  Cz  =  1200  sin  30°. 
Hence  Vx  =  1200  cos  30°,  and  r„  =  -  32«  +  1200  sin  30*.  (3) 


-range  — 
Fig.  181. 


314  ELEMENTARY  FUNCTIONS 

Integrating  (3), 

X  =  1200  cos  SO''-^  +  Cs   and   y=-  \U^  +  1200  sin  30°-«  +  C4. 

To  determine  the  constants  we  note  that  when  i  =  0,  a;  =  0,  ^  =  0,  and 

therefore  Cz  =  0,  and  C4  =  0. 

Hence 

X  =  1200  cos  SO°t     and     y=-  16^  +  1200  sin  aO*'-^.  (4) 

These  equations  give  the  position  of  the  particle  at  any  time.  Solving 
the  first  of  these  equations  for  t  and  substituting  in  the  second  we  find  the 
equation  of  the  path  to  be 

^- -1200^^30°  ^'  +  ^^°^"°-^'  ® 

which  is  the  equation  of  a  parabola. 

To  find  the  range,  let  y  =  0  in  (5)  and  solve  for  x. 

^  1200*  C0S2  30°  tan  30**       00  n^rn  r     i.        ^00       -1 

Hence        x  = —; 38,970  feet  -  7.38  miles. 

16 

At  the  highest  point  Vy  =  0,  hence  from   (3),  0  =  -  32f  +  1200  sin  30°. 

Therefore  the  highest  point  is  reached  when  t  = ^ =  18|  seconds. 

Substituting  this  value  of  t  in  (4),  we  obtain  for  the  maximum  height 
attained 

2/  =  -  16(-V-)'  +  1200  sin  30°-(^''-)  =  5625  feet. 


EXERCISES 

1.  In  the  following  exercises  find  the  equation  of  the  path  in  terms  of  x 
and  .y  by  eliminating  t  from  the  given  equations,  plot  the  path,  find  the 
magnitudes  and  directions  of  v  and  a  for  the  given  value  of  t,  and  at  the 
point  on  the  path  corresponding  to  this  value  draw  lines  representing  v 
and  a  in  magnitude  and  direction. 

(a)  a;  =  2f,  2/  =  f2,  f  =  1.  (b)  ^  =  f,    t/  =  1/t,   t  =  1. 

(c)  x~St^  +  2,  y  2t\  «  -  2.  (d)  a;  =  2t,  y  =  1/(^2  +  i),  ^  .  0. 

(e)  x  =  3<S  y  =  3<,  «  =  1. 

2.  Find  the  equation  of  the  path  of  a  projectile  fired  at  an  angle  of  20® 
to  the  horizontal  with  a  muzzle  velocity  of  1500  feet  per  second;  find  the 
range  of  the  projectile,  the  maximum  height  attained  and  plot  the  path. 

3.  Find  the  equation  of  the  path  of  a  projectile  fired  at  an  angle  of  B° 
to  the  horizontal  with  an  initial  velocity  of  Vq  feet  per  second.  Find  the 
range  and  the  maximum  height  attained.  Find  vq  if  the  range  is  24  miles 
for  d  =  45°. 

4.  A  bomb  is  dropped  from  an  aeroplane  8000  feet  high  moving  hori- 
zontally at  a  velocity  of  120  miles  an  hour.     Determine  how  far  the  bomb 


INTEGRATION  315 

will  fall  from  the  point  on  the  ground  directly  below  the  aeroplane  at  the 
instant  it  was  dropped.  Suggestion:  If  the  point  on  the  ground  is  chosen 
as  the  origjn  with  the  axes  horizontal  and  vertical,  the  initial  conditions 
are  as  given  in  the  table 


i 

X 

y 

Vx 

Vy 

ttx 

ay 

0 

0 

8000 

176 

0 

0 

-32 

5.  A  bomb  is  dropped  from  an  aeroplane  h  feet  high  and  moving  with 
a  velocity  of  Vq  feet  per  second.  If  d  represents  the  distance  from  a  point 
on  the  ground  directly  below  the  aeroplane  when  the  bomb  was  dropped 
to  the  point  where  it  strikes  the  ground,  find  d  as  a  function  of  h  and  vo. 
Plot  the  graph  of  rf  as  a  function  of  Vo,  h  being  constant,  also  of  d  as  a  func- 
tion of  h,  Vo  being  constant.  In  the  first  case  what  is  the  effect  on  d  of 
doubling  Vo,  and  in  the  second  case  of  doubling  h? 

6.  A  ball  rolls  off  a  roof  inchned  at  30°  to  the  horizontal.  If  it  starts 
20  feet  from  the  eaves,  which  are  30  feet  from  the  ground,  where  will  the 
ball  strike  the  ground? 

7.  The  path  of  a  particle  is  given  by  the  equations  x  =  f^  and  y  =  2L 
Plot  the  path,  find  the  acceleration  when  <  =  1,  and  the  components  of 
the  acceleration  along  the  tangent  and  normal  lines  to  the  path  at  the  point 
in  question. 

8.  Find  the  range  of  a  rifle  ball  fired  horizontally  from  a  point  5  feet 
above  the  ground  if  the  initial  velocity  is  2000  feet  per  second.  How  much 
is  the  deviation  from  the  horizontal  at  100  yards?    200  yards? 

9.  A  particle  moves  in  a  circle  whose  center  is  at  the  origin  of  a  rec- 
tangular system  of  axes.  The  angle  (in  radians)  through  which  it  turns 
from  an  initial  position  at  rest  on  the  x-axis  is  given  by  the  equation 
6  =  P  -  t.  Find  the  angular  velocity  co  =  Dt6  and  the  angular  acceleration 
a  =  Dt03  when  t  =  3. 

10.  A  wheel  revolving  at  the  rate  of  120  revolutions  per  second  is  re- 
tarded uniformly  so  that  in  3  seconds,  co  =  90  revolutions  per  second. 
How  long  before  the  wheel  will  come  to  rest,  and  where  will  a  point  P  on 
the  rim  be  if  when  <  =  0  the  angle  for  P  is  ^  =  0? 

109.  Volume  of  a  Right  Prism.  A  'prism  is  a  solid  bounded 
by  two  congruent  polygons  lying  in  parallel  planes  with  their 
corresponding  sides  parallel  and  by  the  parallelograms  de- 
termined by  the  pairs  of  corresponding  sides  of  the  polygons. 
The  other  sides  of  these  parallelograms  are  called  the  lateral 
edges  of  the  prism,  the  polygons  are  called  the  bases j  and  the 
parallelograms  the  lateral  faces  of  the  prism. 

A  right  prism  is  one  whose  lateral  edges  are  perpendicular 
to  the  planes  of  the  bases. 


316 


ELEMENTARY  FUNCTIONS 


/- 

////// 

'///// 

/  /   /    /    /    y  / 

////Ah 

/7 

y 
/ 

Fig.  182. 


A  rectangular  parallelopiped  is  a  right  prism  whose  base  is 
a  rectangle,  e.g.,  an  ordinary  box. 

A  polyhedron  is  a  solid  bounded  by  portions  of  planes.  The 
volume  of  a  polyhedron  is  the  ratio  of  the  polyhedron  to  a  second 
solid  taken  as  the  unit  of  volume.  The  unit  of  volume  is 
usually  a  cube  whose  edge  is  the  Hnear  unit. 

Theorem  1.  The  volume  of  a 
rectangular  parallelopiped  is  the 
product  of  the  area  of  its  base  and 
the  length  of  its  altitude. 

If  the  three  dimensions  of  the 
rectangular  parallelopiped  are  in- 
tegers, h,  j,  k,  the  number  of  unit 
cubes  in  it  is  easily  counted. 
In  any  horizontal  layer  in  the 
figure,  there  are  k  rows  of  cubes,  j  in  each  row.  Hence  by 
the  definition  of  the  multiplication  of  integers,  there  are  jh 
cubes  in  each  layer.  And  since  there  are  h  layers,  the  paral- 
lelopiped contains  hjk  cubes. 

Hence  the  ratio  of  the  parallelopiped  to  the  imit  cube  is  hjkj 
so   that   the  volume   is 
V  =  hjk. 
If  b  is  the  area  of  the 
base,  b  =  jk,  and  hence 
the  volume  V  is 

V  =  hjk^=  bh. 

We  shall  assume  that 
this  result  holds  when 
the  dimensions  are  not 
integers. 

A  block  of  wood  in  the 
form  of  a  rectangular 
parallelopiped  AD'  may 
be  sawed  into  two  congruent  pieces,  as  in  the  figure.     The 


(6) 
FiQ.  183. 


pieces   are 
triangles. 


triangular   right    prisms,   whose    bases   are   right 


INTEGRATION 


317 


Theorem  2.     The  volume  of  a  triangular  right  prism,  whose 

base  is  a  right  triangle,  is  the  product  of  the  base  and  the  altitude. 

If  V  is  the  volume  of  the  triangular  prism  ABCA'B'C,  then 

V   =  i  volume  of  AD' 
=  i  area  ABDC  X  AA' 
=  area  ABC  x  AA' 
^bh, 


and  h  is  the  altitude  of  the 


where  b  is  the  area  of  the  base 

given  triangular  prism. 

Theorem  3.     The  volume  of 

any  triangular  right   prism  is 

the  product  of  the  area  of  the 

base  and  the  length  of  its  alti- 
tude. 
Any  triangular  right  prism 

whose  base  A  BC  is  an  oblique 

triangle  may  be  split  into  two 

triangular  right  prisms  whose 

bases  ADC  and  BDC  are  right  triangles,  as  indicated  in  the 

figure.     Then 

V  =  ABCA'B'C 

=  ADCA'D'C  +  BDCB'D'C 
=  ADC  X  DD'  +  BDC  x  DD' 
=  {ADC  +  BDC)  X  DD' 
=  ABC  X  DD' 
=  bh, 

where  b  is  the  area  of  the  base  and  h  the 

length  of  the  altitude  of  the  given  prism. 
Theorem  4.     The  volume  of  a  right  prism 

is  the  product  of  the  area  of  the  base  and  the 

length  of  its  altitude. 

Let  AD'  be  any  right  prism.     It  may  be 

divided  into  triangular  right  prisms  by  planes 
through  one  of  the  lateral  edges,  say  AA',  and  each  of  the 
non-adjacent  lateral  edges.  If  V  is  the  volume,  6  the  area  of 
the  base,  and  h  the  altitude  of  the  prism,  we  have 


318  ELEMENTARY  FUNCTIONS 

V  =  AD'  =  ABCB'  +  ACDC  -f  ADED'  +  •  •  . 
=  ABC xh+  ACD  xh  +  ADE  x  /H-  •  •  • 
=  {ABC  -f  ACD  +  ADF  +  -  -  -  )h 
=  hh. 

110.  Volume  of  a  Right  Circixlar  Cylinder.  A  cylindrical 
surface  is  a  surface  generated  by  a  straight  Hne  which  moves 
parallel  to  a  given  line  so  as  to  cut  a  given  curve.  The  various 
positions  of  the  moving  line  are  called  the  elements  of  the  surface. 

A  cylinder  is  a  solid  bounded  by  a  cylindrical  surface  and  two 
parallel  planes.  The  sections  of  these  planes  cut  out  by  the 
surface  are  called  the  bases,  and  the  distance  between  them  the 
altitude  of  the  cylinder. 

In  a  right  cylinder  the  elements  are  perpendicular  to  the 
planes  of  the  bases,  and  any  element  is  equal  to  the  altitude. 
If  the  bases  are  circles,  the  solid  is  called  a  right  circular  cylinder. 

Let  r  be  the  radius  of  the  base  and  h  the  altitude  of  a  right 
circular  cylinder. 

Construct  a  right  prism  whose  bases  are  regular  polygons  of 
n  sides  inscribed  in  the  bases  of  the  cyUnder,  and  a  second 

mm 

(a)  

(6) 
Fig.  186. 

prism  whose  bases  are  regular  polygons  of  n  sides  circumscribed 
about  the  bases  of  the  cyUnder.  Let  V  and  V  be  the  volumes 
of  these  prisms,  6'  and  6''  the  areas  of  their  bases.  Their 
altitudes  are  equal  to  h.  Then  by  Theorem  4,  Section  109, 
7'  =  h'h  and  F"  =  h''h. 

Now  let  n  increase  indefinitely.     Then   by  a  theorem  oi^ 
plane  geometry,  6'  and  6"  approach  the  area  of  the  circle 
7rr2,  as  a  limit.     Hence,  by  Theorem  2,  page  266,  V  and  V^ 
approach  the  same  Hmit  Trr%, 


INTEGRATION  319 

This  common  limit  of  V  and  F"  is  defined  to  be  the  volume 
of  the  cylinder,  V.     Hence,  we  have  the 

Theorem.  The  volume  of  a  right  circular  cylinder  is  the  product 
of  the  area  of  its  base  and  the  length  of  its  altitude;  that  is, 

V  =  7rr%. 


EXERCISES 

1.  Find  the  volume  of  a  right  prism  with  a  hexagonal  base  6  inches  on 
a  side  and  an  altitude  of  8  inches.  Find  the  lateral  area  (area  of  the  lateral 
faces)  and  also  the  total  area  of  the  prism. 

2.  Find  the  volume  of  a  cyhnder  if  r  =  16  and  A  =  12. 

3.  Theorem.  If  the  radius  of  a  cylinder  is  r  and  the  altitude  is  h,  then 
the  lateral  area  S  is  S  =  2Trrh,  and  the  total  area  T  is  T  =  2irr{r  +  h). 

Suggestion:  The  curved  surface  of  a  right  cyhnder  may  be  spread  out 
in  the  form  of  a  rectangle. 

4.  A  tin  can  is  3  inches  in  diameter  and  5  inches  high.  Find  the  amount 
of  tin  needed  for  its  construction  and  its  capacity  in  quarts.  (231  cubic 
inches  equal  one  gallon.) 

5.  How  much  space  will  be  occupied  by  a  ton  of  furnace  coal  if  1  cubic 
foot  weighs  52  pounds?  How  many  tons  can  be  placed  in  a  bin  10  by  5  by 
7  feet?    Measure  the  dimensions  of  your  bin  and  calculate  its  capacity. 

6.  How  many  barrels  of  Portland  cement  (weight  =  425  pounds)  will 
be  required  to  construct  a  silo  in  the  form  of  a  hollow  cyhnder  8  inches 
thick,  8  feet  inside  diameter  and  24  feet  high,  if  cement  weighs  90  pounds 
a  cubic  foot? 

7.  What  force  will  be  needed  to  drag  a  block  of  ice  2  by  4  by  6  feet  up 
an  inchne  of  35**,  if  the  density  of  ice  is  0.92  and  1  cubic  foot  of  water 
weighs  62.4  pounds? 

8.  Find  the  area  of  a  cylindrical  water  tank  on  top  of  a  building  if  the 
tank  has  an  altitude  of  6  feet  and  a  diameter  of  5  feet.  What  will  it  cost 
to  Une  it  with  zinc  0.05  of  an  inch  thick,  at  $0.20  a  pound  (density  of  zinc 
=  7.9)?    What  will  be  the  capacity  of  the  tank  in  gallons? 

9.  Find  the  volume  and  total  area  of  a  right  prism  if  its  base  is  a  regu- 
lar pentagon  with  a  side  of  3  inches  and  its  altitude  is  20  inches. 

10.  Find  the  volume  and  total  area  of  the  cyhnder  inscribed  in  the  prism 
in  Exercise  9. 

11.  A  maple  syrup  can  is  a  rectangular  parallelopiped  with  a  square 
base  and  holds  one  gallon.  Find  the  most  economical  proportions  for 
the  manufacturer. 

12.  Find  the  volume  of  the  largest  cyhndrical  package  that  can  be  sent 
by  parcel  post  (see  Exercise  8,  page  288).     (As  a  circle  has  a  greater  area 


320  ELEMENTARY  FUNCTIONS 

than  any  other  plane  figure  with  the  same  perimeter,  a  cylindrical  package 
is  the  largest  package  that  can  be  mailed.) 

13.  Find  the  dimensions  of  a  tomato  can  holding  one  quart,  if  the  amount 
of  tin  used  is  a  minimum  (one  quart  =  57.75  cubic  inches). 

14.  The  length  and  diameter  of  a  cylindrical  bar  are  nearly  24  inches 
and  3  inches  respectively.  Find  the  error  and  relative  error  in  the  volume 
due  to  an  error  of  0.02  of  an  inch  in  measuring  the  diameter. 

15.  What  is  the  allowable  error  in  measuring  (a)  the  diameter  of  the 
base,  4  inches,  (b)  the  height,  12  inches,  of  a  cyhnder  if  its  volume  is  to  be 
determined  within  I  of  one  per  cent?  Assume  in  each  case  that  the  other 
measurement  is  correct.  What  is  the  allowable  percentage  of  error  in  the 
measurement  in  each  case? 

16.  A  piston  sHdes  freely  in  a  circular  cylinder  of  diameter  6  inches. 
At  what  rate  is  the  piston  moving  when  steam  is  admitted  into  the  cylinder 
at  the  rate  of  11  cubic  feet  per  second? 

17.  Assuming  that  a  joint  of  a  bamboo  is  a  cylinder  whose  diameter  is 
proportional  to  its  length,  find  the  rate  of  increase  of  the  volume  in  terms 
of  the  rate  of  increase  of  the  length  when  the  bamboo  is  growing.  Com- 
pare the  rates  when  the  diameter  is  I  inch  and  when  it  is  1|  inches,  assum- 
ing that  the  length  increases  uniformly. 

18.  A  block  of  granite  (density  =  2.7)  3  by  2  by  6  feet  is  being  raised 
by  a  crane.  Find  the  strain  in  the  guy  rope  BC  and  the  thrust  in  the  jib 
AB  as  the  block  is  about  to  swing  to  a  landing.  The  angle  of  elevation 
of  the  jib  is  75°,  and  the  angle  between  the  jib  and  guy  rope,  ABC,  is  10°.2. 

111.  Volume  of  a  Pyramid.  A  pyramid  is  a  solid  bounded 
by  a  polygon  and  a  set  of  triangles  with  a  common  vertex 
whose  bases  are  the  sides  of  the  polygon.  The  polygon  isj 
called  the  base,  the  triangles  the  lateral  faces,  and  the  common 
vertex  the  vertex  of  the  pyramid.  The  altitude  of  the  pyramid 
is  the  perpendicular  distance  from  the  vertex  to  the  base. 

If  the  base  is  a  regular  polygon,  and  if  the  foot  of  the  al- 
titude is  the  center  of  the  base,  the  pyramid  is  said  to  be  regular. 

Theorem  1.  If  a  pyramid  is  cut  by  a  plane  parallel  to  the 
base,  the  section  formed  is  to  the  base  as  the  square  of  its  distance  \ 
from  the  vertex  is  to  the  square  of  the  altitude. 

Let  0-ABCDE  be  the  pyramid,  and  A'B'C'D'E'  the] 
section. 

Let  the  altitude  OP  cut  the  section  at  P'. 

The  pairs  of  triangles  OAB,  OA'B')  OBC,  OB'C;  etc.,  are 
similar  (why?),  and  hence 


INTEGRATION 


321 


A/B[  _0B[  ^  B^  ^  0C_      CD' 
~  OB  "  BC  ~  OC  ^ 


AB 


CD 


(1) 


The  equality  of  the  alternate  ratios  shows  that  the  corre- 
sponding sides  of  the  polygons  are  proportional. 

The  corresponding  angles 
of  the  polygons  are  equal. 
ToshowZ5'C'D'  =  ZJ5CD, 
for  example,  draw  BD  and 
B'D\  Then  since  the  tri- 
angles OBD  and  OB'D'  are 
similar  (why?) 


OB' 
OB 


B'D' 
BD' 


and  by  combining  this  result 
with  (1)  it  is  seen  that  the 
corresponding  sides  of  the 
triangles  BCD  and  B'C'D' 
are  proportional.  Hence 
these  triangles  are  similar, 
and  therefore  Z  B'C'D' 
=  ZBCD. 

Since  the  corresponding 
angles  of  the  polygons  are 
equal,  and  the  correspond- 
ing sides  proportional,  the 
polygons  are  similar. 

Hence  their  areas  are  proportional  to  the  squares  of  any 
two  corresponding  sides,  so  that 

A'B'C'D'E'      A'B'^ 


Fig.  187. 


ABODE 


AB' 


But  from  (1),  and  from  the  similar  triangles  OBP  and  OB'P', 

A'B'      OB'      OP' 
AB      OB       op' 
A'B'^     OP'^ 


Hence 


AB^       OP^' 


322 


ELEMENTARY  FUNCTIONS 


so  that, 


A'B'C'D'E'      qP2 
ABCDE     ~  OF"  ' 


Theorem  2.  The  volume  of  a  pyramid  is  one-third  the  product 
of  the  area  of  the  base  by  the  length  of  the  altitude. 

Let  V  denote  the  volume 
of  that  part  of  the  pyramid 
between  the  vertex  and  a 
section  parallel  to  the  base 
at  a  distance  x  from  the 
vertex. 

Let  b  be  the  area  of  the 
base,  6'  the  area  of  the 
section,  and  h  the  alti- 
tude. 

Let  b"  be  the  area  of  a 
section  parallel  to  the  base 
at  a  distance  x  +  Ao:  from 
the  vertex. 

When  X  increases  by  Ax, 
V  increases  by  an  amount 
AF  equal  to  the  part  of 
the  pyramid  between  V 
and  b".  This  part  of  the 
pyramid  is  greater  than 
a  prism  with  base  b'  and 
altitude  Ax,  and  less  than 
and  altitude  Ax. 


Fig.  188. 


a  prism  with  base  V 

Hence  6'Ax<AF<6"Ax 

Dividing  by  Ax  we  have 


A7 


As  Ax  approaches  zero,  6"  approaches  b'  and  -r—  approaches  DxV. 
Hence,  passing  to  the  limit, 

Z>,y-6'.  (1) 


I 


INTEGRATION 

But  by  Theorem  1, 

b 

=  |or6'. 

-1- 

Hence,  substituting 

in 

(1),      D.V 

-i.-- 

Integrating 

V 

-       ^    X3 

+  c. 

As  7  =  0  when  x  = 

=  0, 

we  have  C 

=  0,  and  hence 

x3. 

323 


When  X  =  h,  the  value  of  V  is  the  volume  of  the  given  pyramid. 
Hence  the  required  volume  is 

V  =—  h^  =  -bh 
^      3/i2  ^       S^''- 

112.  Volume  of  a  Solid  of  Revolution.  A  solid  of  revolution 
is  a  solid  generated  by  revolving  an  area  about  an  axis.  Such 
a  solid  may  be  turned  out  on  a  lathe.  A  section  of  the  solid 
by  a  plane  perpendicular  to  the  axis  is  always  a  circle. 

Theorem.  The  volume  V  generated  by  revolving  about  the 
X-axis  the  area  bounded  by  the  curve  y  =  f{x),  the  x-axis,  and  the 
ordinates  to  the  curve  at  the  points  whose  abscissas  are  a  and 
X  is  such  that 

D^V  =  7ry2. 

Let  the  soUd  V  be  bounded  on  the  left  by  the  plane  per- 
pendicular to  the  a;-axis  at  A  (a,  o).  The  radius  of  the  circle 
bounding  V  on  the  right  is  y  =  f{x).  Denote  the  area  of  this 
section  by  6'  (Fig.  189). 

If  X  increases  by  Aa;,  V  increases  by  an  amount  AT  which  is 
the  volume  of  that  part  of  the  solid  lying  between  V  and  a 
section  b"  at  a  distance  Lx  from  b\  If  y  increases  as  x  in- 
creases, then  b'<b'\  and  AV  is  greater  than  a  cylinder  with 
base  b'  and  altitude  Ax,  and  less  than  a  cylinder  with  base  6" 
and  altitude  Ax.    Hence: 

b'AX<AV<b''Ax. 


324  ELEMENTARY  FUNCTIONS 

Dividing  by  Ax, 


Ay 


As  Ax  approaches  zero,  V  approaches  V  and  -r—  approaches 

D^V.    Therefore 
D,V  =  b\ 
Hence, 

The  volume  of  the 
soUd  between  the 
planes  perpendicular 
to  OX  at  A  and  a 
second  given  point 
B{b,  o)  is  found  by 
integrating,  setting 
y  =  0  and  X  =  a  to 


Fig.  189. 

determine  the  constant  of  integration,  and  then  setting  x  =  h. 


Example.  Find  the  volume  of  the  soUd  generated  by  revolving  about 
the  a;-axis  the  area  bounded  by  z/  =  -y/x,  the  x-axis,  and  the  ordinates  at 
X  =  1  and  a:  =  4. 

Let  V  denote  the  volume  between  the  planes  perpendicular  to  the  x-axig 
at  X  =  1  and  the  point  whose  abscissa  is  x.     Then  by  the  theorem 


Integrating, 


Since  7  =  0  when  x  =  1, 


0  =  2  +  C',  so  that  C 


Hence 


Fig.  190. 


The  required  volume  is  found  by  setting  x  =  4,  which  gives 


INTEGRATION 


325 


113.  Volume  of  a  Cone  of  Revolution.  A  cone  of  revo- 
lution is  a  cone  that  may  be  generated  by  revolving  a  right 
triangle  about  one  leg  as  an  axis. 

Theorem.  The  volume  of  a  cone  of  revolution  is  one  third  the 
product  of  the  area  of  the  base  and  the  attitude. 

Consider  the  cone  in  the  figure  as  generated  by  revolving 
the  triangle  OAB  about  OA.  Take  the  origin  at  0,  and  the 
X-axis  along  OA. 

Let  r  =  AB  denote  the  radius  of  the  base  of  the  cone,  and 
h  =  OA  the  altitude. 

The  slope  of  OB  ism  =  AB/OA  =  r/h. 

And  hence  the  equation  of  OB  is 


y  =  mx  =  -rX' 
n 


Therefore  if  V  denotes  the  volume  from  the  vertex  0  to  a 
section  at  a  distance  x  from  0 
(Theorem,  Section  112), 


D,V  =  Try' 
Integrating 


h^^ 


As   V  =  0   when   x  =  0,   then 


C  =  0,  and  hence  V 


Fig.  191. 


Zh' 


The  volume  of  the  given  cone  is  obtained  when  x  =  h.    Re- 
membering that  the  area  of  the  base  h  =  Trr^,  we  have 


V^'^h'' 
Sh^ 


-^Trr% 

o 


\in. 


EXERCISES 

1.  Find  the  volume  of  a  regular  octagonal  pyramid  if  a  side  of  the  base 
is  4  inches  and  a  lateral  edge  is  6  inches. 

2.  Find  the  volume  of  a  cone  of  revolution  if  the  diameter  of  the  base  is 
12  inches  and  the  altitude  is  6  inches. 


326  ELEMENTARY  FUNCTIONS 

Definition.  A  straight  line  drawn  from  the  vertex  of  a  cone  of  revo- 
lution to  a  point  in  the  circumference  of  the  base  is  called  the  slant  height 
of  the  cone.  It  is  one  of  the  positions  of  the  hypotenuse  of  the  generating 
triangle. 

3.  Find  the  volume  of  a  cone  of  revolution  if  the  slant  height  is  13 
inches  and  the  diameter  of  the  base  is  10  inches. 

Definition.  The  slant  height  of  a  regular  pyramid  is  the  altitude  of 
one  of  the  equal  isosceles  triangles  forming  its  lateral  faces. 

4.  Find  the  lateral  area  and  the  total  area  of  the  pyramid  in  Exercise  1. 

5.  Find  the  lateral  and  total  area  of  the  cone  in  Exercise  2. 

Hint:  The  lateral  surface  of  a  cone  of  revolution  may  be  spread  out  in 
the  form  of  a  sector  of  a  circle. 

6.  Theorem.  The  lateral  area  of  (a)  a  regular  pyramid,  (6)  a  cone  of 
revolution,  is  half  the  product  of  the  perimeter  of  the  base  and  the  slant 
height. 

7.  Find  the  volume  generated  by  revolving  about  the  a:-axis  the  area 
bounded  by  the  hnes  y  =  3rc,  y  =  0,  and  x  =  3.  Check  the  result  by  the 
theorem  in  the  preceding  section. 

8.  State  and  prove  a  theorem  analogous  to  that  on  page  323  for  the 
volume  of  a  solid  generated  by  revolving  an  area  about  the  y-axis. 

9.  Find  the  volume  generated  by  revolving  about  each  axis  the  area 
boimded  by  the  parabola  y  =  4^  -  x^  and  the  positive  parts  of  the  axes. 

10.  Find  the  volume  generated  by  revolving  about  the  x-axis  the  area 
bounded  by  the  parabola  ?/  =  -  x^  +  4tx  -3  and  the  x-axis. 

11.  Find  the  volume  of  the  sphere  generated  by  revolving  the  circle 
x2  -f.  ?/2  =  16  about  the  x-axis. 

12.  Find  the  volume  generated  by  revolving  about  the  y-axis  the  area 
bounded  by  y  =  x^,  x  =  0,  and  y  =  9. 

13.  Find  the  volume  of  the  sohd  generated  by  revolving  about  the  x-axis 
the  area  bounded  by  y  =  \x  +  2,  the  x-axis,  and  the  ordinates  at  x  =  2 
and  X  =  6. 

Definition.  A  frustum  of  a  cone  is  that  part  of  a  cone  included  be- 
tween two  planes  perpendicular  to  the  axis.  It  may  be  generated  by  re- 
volving a  trapezoid  ABCD,  such  that  AD  and  BC  are  perpendicular  to 
AB,  about  AB.    See,  for  example,  the  solid  obtained  in  Exercise  13. 

14.  If  that  part  of  the  hne  y  =  re  +  2  which  lies  between  x  -  1  and 
X  =  4  is  revolved  about  the  x-axis,  find  the  volume  of  the  frustum  of  a  cone 
which  is  generated. 

15.  Find  the  volume  of  a  frustum  of  a  cone  if  the  radii  of  the  bases  are 
3  and  7  inches  and  the  altitude  is  19  inches. 

16.  The  altitude  of  a  conical  receptacle  is  10  inches,  and  the  radius  of 
the  base  is  6  inches.  The  axis  is  vertical  with  the  vertex  at  the  bottom. 
If  water  is  poured  into  it  at  the  rate  of  20  cubic  inches  per  second,  how 
fast  is  the  surface  rising  when  the  depth  is  5  inches? 


INTEGRATION 


327 


17.  The  water  reservoir  of  a  town  is  in  the  form  of  an  inverted  conical 
frustum  with  the  sides  inclined  at  an  angle  of  45°,  the  radius  of  the  smaller 
base  being  100  feet.  If  the  depth  is  decreasing  at  the  rate  of  5  feet  per 
day  when  the  water  is  20  feet  deep,  show  that  the  town  is  being  supphed 
with  water  at  the  rate  of  72,000  tt  cubic  feet  per  day. 

18.  The  diameter  of  the  base  of  a  cone  is  found  to  be  6  inches  and  the 
altitude  15  inches.  If  the  error  in  each  measurement  is  not  greater  than 
one  per  cent,  find  the  relative  error  in  the  computed  volume,  assuming 
first  that  the  measurement  of  the  diameter  is  exact,  and  then  that  the 
measurement  of  the  altitude  is  exact. 

19.  The  diameter  of  the  base  of  a  cone  is  found  to  be  8  inches  and  the 
slant  height  10  inches.  Assuming  that  the  first  measurement  is  exact, 
what  is  the  admissible  error  in  the  second  if  the  volume  is  to  be  determined 
within  0.1  of  one  per  cent? 

20.  How  many  cubic  feet  of  air  will  there  be  in  the  largest  conical  tent 
that  can  be  made  out  of  200  square  feet  of  canvas? 

21.  What  is  the  least  amount  of  canvas  that  can  be  used  to  make  a 
conical  tent  of  1200  cubic  feet  capacity? 

114.  Volume  of  the  Sphere.  A  sphere  may  be  generated 
by  revolving  a  circle  about  a  diameter.  Choosing  the  center 
of  the  circle  as  origin,  a  point  Pix,  y)  will  He  on  the  circle  if 
and  only  if  OP  =  r,  where  r  is  the 
radius. 

From  the  right  triangle  OMPj 

X2  +  1/2    =   OP^, 

and   hence   the   equation  of    the    - 
circle  is 

Let  this  circle  be  revolved  about 
the  X-axis,  and  denote  the  ex- 
tremities of  the  diameter  on  the 
rr-axis  by  A(-  r,  0)  and  B{r,  0). 

Let  V  denote  the  volmne  from  A  to  a  section  cutting  the 
X-axis  at  a  point  whose  abscissa  is  x. 

By  the  theorem  on  page  323,  we  have,  from  (1), 

Z)xF  =  7ri/2  =  7r(r2  _  ^2), 
Integrating 


Fig.  192. 


328  ELEMENTARY  FUNCTIONS 

Since  7=0  when  x  =  -  r, 

0  =  7r(-  r3  +  »^/3)  +  C,    whence    C  =  27rr3/3. 
Hence    . 

V  =  Tr{r^x  -  a^/S)  +  2Tr^/S, 

When  X  =  r,  V  is  the  volume  of  the  sphere,  hence  the  volume 
of  the  sphere  is 

V  =  7r(r3-  ^/3)  ^  27rr3/3  =  iwr^. 

Hence  the 

Theorem,     The  volume  of  a  sphere  of  radius  r  is  V  =  iirr^. 

115.  Area  of  a  Sphere.  Imagine  a  very  large  number  of 
planes  drawn  tangent  to  the  sphere.  The  area  and  volume 
of  the  polyhedron  bounded  by  these  planes  would  be  very 
nearly  equal  to  the  area  and  volume  of  the  sphere.  Planes 
passed  through  the  center  of  the  sphere  and  the  lines  of  in- 
tersection of  the  tangent  planes  would  divide  the  polyhedron 
into  as  many  Uttle  pyramids  as  there  are  tangent  planes. 

These  pjrramids  would  all  have  the  radius  for  altitude,  and 
the  simi  of  their  base  areas  would  be  the  area  of  the  polyhedron. 

Hence  the  volume  of  the  polyhedron  would  be  one-third  its 
area  multiplied  by  the  radius. 

This  would  be  true  no  matter  how  large  the  number  of  faces 
of  the  polyhedron.  If  the  number  of  tangent  planes  is  in- 
creased indefinitely  in  such  a  way  that  the  area  of  the  base  of 
each  little  pyramid  approaches  zero,  then  the  volume  and 
area  of  the  polyhedron  approach  respectively  the  volume  and 
area  of  the  sphere. 

Hence  the  volume  of  the  sphere  is  also  equal  to  one-third 
its  area  multiplied  by  the  radius. 

Denoting  the  area  of  the  sphere  by  *S  and  applying  the 
theorem  in  the  preceding  section,  we  have 

iSr  =  jTrr^,    whence    S  =  4irr^. 
Hence  the 

Theorem.     The  area  of  a  sphere  of  radium  risS  =  ^wr^. 


INTEGRATION  329 

EXERCISES 

1.  Find  the  area  and  volume  of  a  sphere  whose  radius  is  5  inches. 

2.  Find  the  volume  of  the  sphere  inscribed  in  a  cone  of  revolution  whose 
altitude  is  12  inches,  the  radius  of  whose  base  is  5  inches. 

3.  For  what  values  of  the  radius  is  the  number  of  units  in  the  volume  of 
a  sphere  less  than  the  number  of  units  in  the  area?  For  what  values  of 
the  radius  is  DtV<DtS?  DtV>DS?  In  biology,  we  learn  that  a  cell 
subdivides  after  a  certain  time.    Why? 

4.  The  diameter  of  a  sphere  is  found  by  measurement  to  be  8.3  inches 
with  a  probable  error  of  0.1  inch.  What  is  the  error  in  the  computed 
value  of  the  area  of  the  sphere?  In  the  computed  value  of  the  volume?  Is 
the  percentage  error  in  the  area  greater  or  less  than  the  percentage  error 
in  the  volume? 

5.  What  is  the  relation  between  the  rates  of  increase  of  the  radius  and 
volume  of  a  soap  bubble?  When  the  radius  is  3  inches  and  is  increasing 
at  the  rate  of  0.1  inch  per  second,  how  fast  is  the  volume  increasing?  The 
surface? 

6.  A  washbowl  is  in  the  form  of  a  hemisphere  of  radius  8  inches.  How 
much  water  is  there  in  it  when  the  water  is  5  inches  deep? 

7.  When  the  depth  of  the  water  in  the  bowl  of  Exercise  6  is  4  inches  the 
surface  is  f alHng  at  the  rate  of  |  of  an  inch  per  second.  How  rapidly  does 
the  water  run  off  through  the  drain  pipe? 

8.  Find  the  area  and  volume  of  the  earth,  assuming  that  it  is  a  sphere 
whose  radius  is  3957  miles.  Find  the  error  and  relative  error  in  each  case 
if  the  error  in  the  radius  is  not  more  than  7  miles.  How  does  the  relative 
error  in  the  area  and  in  the  volume  compare  with  the  relative  error  in  the 
radius? 

9.  How  many  lead  shot  i  of  an  inch  in  diameter  can  be  made  from  a 
piece  of  lead  pipe  2  feet  long  whose  outside  and  inside  diameters  are  re- 
spectively 1.25  inches  and  1  inch? 

MISCELLANEOUS   EXERCISES 

1.  Find  the  volume  generated  by  revolving  the  area  bounded  by  the 
curves  y  =  x^  and  x  =  y^  about  the  rc-axis. 

2.  A  ball  rolls  down  a  smooth  plane  inchned  at  an  angle  of  18°.  Find 
how  far  it  will  roll  in  t  seconds. 

3.  The  diameter  of  a  cone  is  found  to  be  5  inches,  and  the  altitude  8 
inches.  If  the  error  in  the  diameter  is  0.06  of  an  inch  and  the  altitude  is 
exact,  find  the  error  in  the  computed  value  of  the  lateral  area. 

Definition.  At  the  center  0  of  an  equilateral  triangle  ABC  erect  a 
line  perpendicular  to  the  plane  of  the  triangle.  Take  a  point  D  on  it  such 
that  AD  =  AB,  and  join  D  to  A,  B,  and  C.  The  figure  so  obtained  is 
called  a  regular  tetrahedron  [Fig.  193  (a)]. 


330 


ELEMENTARY  FUNCTIONS 


4,  Find  the  altitude  DO  and  the  volume  of  a  regular  tetrahedron 
(a)  whose  edge  AB  is  6  inches;   (b)  whose  edge  is  e. 

Definition.  At  the  center  0  of  a  square  ABCD  erect  a  hne  perpen- 
dicular to  the  plane  of  the  square,  and  extend  it  on  both  sides  of  the  plane 
to  points  E  and  F  such  that  EA  =-  FA  =  AB.  Join  E  and  F  to  the  vertices 
of  the  square.    The  resulting  figure  is  called  a  regular  octahedron. 


Fio.  193. 


6.  Find  the  volume  of  a  regular  octahedron  (a)  whose  edge  ^iS  is  6 
inches;  (b)  whose  edge  is  e.  (Note  that  EF  is  a  diagonal  of  the  square 
AECF.) 

6.  Express  the  total  area  of'  the  octahedron  in  terms  of  the  edge.  Do 
the  same  for  the  cube  and  the  regular  tetrahedron,  and  plot  the  graphs  of 
these  three  functions  of  e  on  the  same  axes.  For  the  same  value  of  e, 
which  soUd  has  the  greatest  area?  Which  area  increases  the  most  rapidly 
as  e  increases? 

7.  The  graph  of  xi  +  yi  "  ai  is  a  parabola  tangent  to  the  coordinate 
axes  whose  axis  of  symmetry  bisects  the  first  quadrant.  Find  the  area 
bounded  by  the  curve  and  the  axes. 

8.  Find  the  volume  of  the  solid  generated  by  revolving  about  the  y-axis 
the  area  bounded  by  the  parabola  y  =  ax^,  x  =  0,  and  y  =  h.  Show  that 
this  volume  is  one-half  the  volume  of  a  cylinder  whose  altitude  is  h  and 
whose  bases  are  equal  to  the  circle  forming  the  upper  surface  of  the  solid. 

9.  A  point  moves  so  that  its  coordinates  at  any  time  t  are  given  by 


2t 

1  +  fi' 


1  -t\ 
1  + 1^' 


Find  the  components  parallel  to  the  axes  of  the  velocity  and  acceleration. 
Show  that  the  point  moves  in  a  circle  (to  eliminate  t,  square  both  equations 


INTEGRATION  331 

and  add  the  results)  and  determine  the  part  of  the  circle  it  describes  from 
t  =  0  to  t  =  1.  Find  the  velocity  and  acceleration  (direction  and  magni- 
tude) at  these  times 

10.  A  water  resevoir  is  in  the  form  of  the  figure  generated  by  revolving 
the  curve  12y  =  x*  about  the  y-axis.  How  much  does  it  contain  if  the 
water  is  10  feet  deep  at  the  center?  If  water  is  entering  at  the  rate  of 
10  cubic  feet  per  second  when  the  depth  at  the  center  is  12  feet,  how  rapidly 
is  the  surface  rising? 


CHAPTER  VIII 
PROPERTIES   OF  TRIGONOMETRIC   FUNCTIONS 

LOGARITHMIC  SOLUTION  OF  TRIANGLES,  CASES  m  AND  IV 

116.  Introduction.  In  this  Chapter  we  shall  derive  formulas 
which  express  properties  of  the  trigonometric  functions  anal- 
ogous to  certain  properties  of  algebraic,  exponential,  and 
logarithmic  functions  with  which  we  are  already  familiar. 
These  properties  enable  us  to  change  the  form  of  expressions 
involving  trigonometric  functions,  a  process  of  great  importance 
in  those  parts  of  more  advanced  mathematics  where  these  func- 
tions appear.  These  properties,  together  with  the  formulas  for 
the  functions  of  n90°  =*=  d  (page  192),  are  used  more,  perhaps, 
than  the  solution  of  triangles,  except  in  such  fields  as  surveying. 

As  an  application  of  these  formulas  we  shall  obtain  formulas 
for  the  solution  of  triangles,  which  are  adapted  to  the  use  of 
logarithmic  tables,  for  those  cases  in  which  we  have  hitherto 
used  the  law  of  cosines. 

In  the  sections  immediately  following  we  shall  consider  the 
interdependence  of  the  trigonometric  functions  of  an  angle  6, 
with  applications  to  the  solution  of  equations  and  the  proof  of 
identities. 

117.  Fundamental  Trigonometric  Relations.  As  an  im- 
mediate consequence  of  the  definitions  of  the  trigonometric 
functions  we  obtained  the  reciprocal  relations 

cote  =  ^^;  (1) 

CSC  e  =  -j^-  (3) 

sin  (7 

332 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       333 


In  the  same  section,  we  saw  that 

.       /,      sin  ^ 
tan  6  = 


(4) 


cos  6 
To  these  relations  we  now  add 

sin2  d  =  cos2  ^  =  1,  (5) 

in  which  sin^^  denotes  the  square  of  sin  d,  or  (sin  dy,  and  cos^  6 
denotes  (cos  dy. 
To  prove  (5),  we  have,  for  any  value  of  d, 


sin2  6  +  cos2  6 


2/2         X^ 
-2   +72 


y^  +  x' 


These  five  formulas  are  known  as 
the  fundamental  formulas  of  trigo- 
nometry. They  are  independent,  that 
is,  no  one  of  them  is  a  consequence 
of  the  other  four,  and  they  may  be 
solved  for  any  five  of  the  six  func- 
tions in  terms  of  the  sixth.  This  is 
illustrated  in  the 


Fig.  194. 


Example.    Express  each  of  the  trigonometric  functions  in  terms  of  sin  d. 
From  (5)  we  obtain  cos  0  =  ±\/l  -  sin^  6. 

sin  6 


Substituting  in   (4), 
Then  by  (1),  (2),  (3), 


tan0  = 


VI  -  sin2  e 


cot  ty  =  : — ^ ,  sec  ^  = 

sm  6        ' 


CSC  6 


1 

sin  0 


VI  -  sin2  0 

The  sign  of  the  radical  must 
be  chosen  from  a  knowledge  of 
the  quadrant  in  which  0  lies. 

These  results  may  also  be 
obtained  from  a  figure  by 
means  of  the  definitions  of  the 
functions  (see  Exercises  9  and 
10,  page  170).  The  given 
function,  sin  0,  may  be  re- 
garded as  a  fraction  with  unit 
denominator.  Describe  a  circle 
about  the  origin  with  radius  r  =  1,  and  draw  the  line  y  =  sin  0,  regarding 
0  as  constant.    Let  the  line  cut  the  circle  in  P  and  P'.    Then  the  angles 


334  ELEMENTARY  FUNCTIONS 

XOP  and  XOP'  are  such  that  their  sines  are  equal  to  ^^,  and  hence 

these  angles  are  values  of  6. 

From  the  right  triangles,  using  the  Pythagorean  Theorem,  we  get 


OM  =  VI  -  sin2  e        and        OM'  =  -y/l  -  sin^  d. 

Then  the  results  obtained  above  may  be  written  down  from  the  figure. 
In  this  figure  it  is  assumed  that  sin  6  is  positive.  If  sin  6  is  negative,  the 
angles  would  he  in  the  third  and  fourth  quadrants. 

This  example  shows  that  it  would  be  entirely  possible  to  get 
along  with  only  one  of  the  trigonometric  functions,  but  fre- 
quently it  would  be  inconvenient. 

The  formulas  (l)-(5)  are  identities  (Definition,  page  134). 
There  can  not  be  more  than  five  independent  equations  con- 
necting the  six  functions,  for  if  there  were  six,  it  would  be 
possible  to  solve  them  for  the  six  functions,  which  would  then 
be  constants.  Hence  any  other  identity  involving  functions 
of  a  single  angle  6  must  be  a  consequence  of  these  five,  and 
may  be  proved  by  them. 

•   Three  other  identities  are  of  sufficient  importance  to  be  listed 
with  the  fundamental  formulas.     They  are: 

J.  a     cos  ^ 
cot  6  =  ^—A'  (6) 

l  +  tan2  0  =  sec2^.  (7) 

1  +  cot2  d  =  csc2  e.  (8) 

The  first  two  may  be  proved  as  follows,  the  proof  of  (8) 
being  analogous  to  that  of  (7). 

1  1         cos  ^ 


cot^ 


tan  6      sin  6      sin  6 
cos  6 


1  +  tan^  ^  =  1  4-  — 2-2  = 2Q =  — rh  =  sec^ 6. 

cos^  0  cos^  a  cos^  0 

The  formulas  in  this  and  the  following  sections  should  be 
memorized. 

118.  Trigonometric  EquatioQS.  No  specific  rules  for  the 
solution  of  equations  involving  trigonometric  functions  can 
be  given,  but  the  following  remarks  may  be  of  service. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      335 

Express  the  equation  in  terms  of  a  single  function  by  means 
of  the  formulas  in  the  preceding  section,  and  solve  for  this  function. 
The  values  of  the  angle  may  then  be  found  from  the  tables. 

//,  when  all  the  terms  are  written  on  the  left  of  the  equality 
sign,  the  left-hand  member  can  be  factored,  equate  each  factor  to 
zero,  and  solve  the  resulting  equations.  For  any  solution  makes 
the  product  of  the  factors  zero,  and  hence  it  must  make  one 
of  the  factors  zero.  Conversely,  any  solution  of  one  of  the 
resulting  equations  makes  one  of  the  factors  zero,  and  hence 
also  the  product. 

Example  1.     Solve  tan  a;  -  2  sin  x  =  0. 

By   (4),  Section   117,  ^-^^-^  -  2  sin  x  =  0. 
•^    ^    '  '  cos  X 

Multiplying  by  cos  x,  sin  a;  -  2  sin  x  cos  x  =  0. 
Factoring,  sin  x  (1  -  2  cos  x)  =  0. 

Equating  each  factor  to  zero  and  solving, 

sin  X  =  0,  or        cos  x  =  \. 

Hence  a;  =  0,  tt        or  x  =  7r/3,  57r/3. 

All  the  other  values  of  x  may  be  obtained  by  adding  2mr  to  each  of  these 
four  solutions. 

To  check  the  results,  we  substitute  each  of  the  values  of  x  in  the  given 
equation.    This  gives: 

If  X  =  0,  tan  X  -  2  sin  X  =  0  -  2  X  0  =  0. 

If  X  =  -TT,  tan  X  -  2  sin  X  =  0  -  2  X  0  =  0. 

If  X  =  7r/3,  tan  X  -  2  sin  X  =  V3  -  2  X  V3/2=  0. 

If  X  =  57r/3,  tan  X  -  2  sin  X  =  -  VS  -  2(-  V3/2)  =  0. 

Hence  the  results  obtained  are  correct. 

Example  2.     Solve  the  equation  2  sin  x  -  cos  x  =  1. 

By  (5),  Section  117,  2  sin  x  =fc  Vl  -  sin^  x  =  1. 

Transposing,  2  sin  x  -  1  =  ±\/l  -  sin^x 

Squaring   both   sides,    4  sin^  x-4sinx  +  l  =  l-  sin^  x, 
whence  5  sin^  x  -  4  sin  x  =  0. 

Factoring,  sin  x  (5  sin  x  -  4)  =  0, 

so  that  sin  X  =  0    or    sin  x  =  |  =  0.8, 

and  henc^  x  =  0  or  tt,  or  x  -  53°.  13  =  0.9272  or  126.'*87  =  2.2144. 


336  ELEMENTARY  FUNCTIONS 

The  values  of  the  last  two  angles  in  radians  are  obtained  from  page  32 
of  the  Tables. 

It  is  essential  that  these  results  be  checked,  as  extraneous  roots  are 
frequently  introduced  when  an  equation  is  squared.    We  have: 

lfa;  =  0,  2sinx-cosa;  =  2x0-l  =  -l. 

If  X  =  TT,  2  sin  x  -  cos  X  =  2  X  0  -  (-  1)  =  1. 

Jfx^  0.9272,  2  sin  a;  -  cos  X  =  2  X  0.8  -  0.6  =  1. 

If  X  =  2.2144,  2  sin  0?  -  cos  X  =  2  X  0.8  -  (-  0.6)  =  2.2. 

The  first  and  last  values  of  x  do  not  satisfy  the  given  equation,  and 
they  are  therefore  discarded.  The  values  x  =  ir  and  x  =  0.9272  do 
satisfy  the  equation.  All  other  solutions  may  be  obtained  from  these 
by  adding  2n7r. 

119.  Trigonometric  Identities.  Identities  constitute  an  im- 
portant part  of  mathematics.  If  we  encounter  a  complicated 
fraction  in  the  solution  of  a  problem,  we  proceed  to  simplify  it. 
The  reduction  of  the  fraction  is  essentially  a  proof  of  the 
identity  obtained  by  equating  the  original  fraction  to  the 
simpler  result  which  is  foimd. 

The  logarithmic  identity, 

,      a  sin  B      ,  i       •     r>      ^       •     a 

log — : — J-  =  log  a  +  log  sm  B  -  log  sm  A, 

which  is  proved  by  Theorems  7  and  8,  page  223,  is  the  basis  of 
computing  b,  sl  side  of  a  triangle,  if  a,  A,  and  B  are  given. 

Trigonometric  identities  are  of  the  same  importance  in 
dealing  with  expressions  involving  the  trigonometric  functions 
as  algebraic  and  logarithmic  identities  are  in  working  with 
algebraic  and  logarithmic  functions. 

Trigonometric  identities  involving  functions  of  a  single 
angle  may  be  proved  by  means  of  the  fundamental  formulas  in 
Section  117  and  the  operations  of  algebra,  by  transforming  one 
member  of  the  identity  into  the  other.  The  proofs  of  equa- 
tions (6)  and  (7)  in  the  section  cited  are  illustrations  of  the 
method. 

Example  1.    Prove  the  identity, 

1 


sec*  X  -  sec  a;  tan  z 


1  +sina; 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      337 

By  the  formulas  in  Section  117  we  obtain 
sec*x 


Example  2.       Prove 
Proof. 


1         1 

sec  z  tan  z  =  — r~ 

cos2  z     cos  z 

sinx 

C08X 

1  -  sin  x 

COS^  X 

1  -  sin  X 

~  1  -  sin  *  a; 

1 

1  +  sinx 

sin  a;          1  +  cos  x 

1  -  cos  X  ""      sin  X 

sin  X               sin  x  (1  +  cos  x) 

1  -  cos  X  "  (1  -  cos  x)  (1  +  cos  x) 

sin  X  (1  +  cos 

1  -  COS^  X 

sin  X  (1  +  cos  x) 
sin^  X 

1  +  cos  X 

Another  method  of  establishing  the  truth  of  an  identity  is 
to  show  that  both  members  can  be  reduced  to  the  same  form. 
Still  another  is  to  transform  the  given  equation  until  it  is 
reduced  to  some  known  identity. 

EXERCISES 

1.  Express  formulas  (l)-(8),  Section  117,  in  words. 

2.  By  means  of  the  formulas  in  Section  117,  find  all  the  functions  of  0, 
given 

(a)  cos  e  =  j%.  (b)  tan  d  =  1  (c)  esc  ^  =  3. 

Check  the  results  by  the  method  used  in  Exercise  10,  page  170. 

3.  Using  both  methods  in  the  Example  in  Section  117,  express  each  of 
the  functions  in  terms  of 

(a)  cos  0.  (b)  tan  0.  (c)  cot  0.  (d)  sec  0.  (e)  esc  0. 

4.  Express 

,  V  tan  0  +  cot  0  .    .  e   •    o      J        a 

W  ~T~5 — I a  111  terms  of  sm  u  and  cos  u. 

^      cot  0  -  tan  0 

,,  V  sec  0  -  tan^  0  .    .  .        n 

(b)  r-T-fl 111  terms  of  cos  0. 

sm^  t7 

,  V     2  sin  0  cos  ^   .     .  t  i.      a 

(c)  — r-5 ^-r-a  in  terms  of  tan  9. 

cos''  u  -  sm^  c' 


338  ELEMENTARY  n^NCTIONS 

5.  Solve  the  equations: 

^  (a)  2  sin2  a:  +  3  cos  a;  =  0. 

(b)  cot  X  -2  cos  a:  =  0. 

(c)  sin  X  tan  x  -  cos  a;  =  0. 

(d)  sin  X  +  cos  x  =  \. 

(e)  sin  X  +  3  cos  a;  =  1. 

(f)  sin  a;  +  3  cos  X  =  2. 

6.  Prove  the  identities: 

(a)  sin2  Q  -  cos^  ^  =  2  sin^  ^  -  1. 

(b)  2  sin2  ^-1  =  1-2  cos^  B, 

(c)  tan  0  +  cot  0  =  sec  Q  esc  ^. 
1  -  sin  a:  cos  x 


(d) 


cos  a;  1  -  sin  a; 

1 


(e)  csc2  0  +  cot  0  esc  ^  =  -  ^ 

1  —  cos  a 

(f)  (2  sin  X  cos  a;)*  +  (cos^  x  -  sin^  a:)^  =  1. 

120.  Functions  of  the  Sum  of  Two  Angles.    The  relatioD 

(a  +  6)2  =  a2  4-  2a&  +  h^ 

expresses  in  a  different  foim  the  square  of  the  sum  of  two  num- 
bers. We  seek  now  an  analogous  expression  for  the  sine  of 
the  sum  of  two  angles,  sin  (0  +  <^),  where  ^  and  </>  are  any  two 
acute  angles.  The  sum  ^  +  </>  is  constructed  by  using  the 
terminal  line  of  0  as  the  initial  line  of  <^.  It  may  be  either  an 
acute  or  an  obtuse  angle,  as  indicated  in  the  figures. 

Let  A  be  any  point  on  the  terminal  line  of  ^,  and  draw  AB 
perpendicular  to  0  X.    Then 

8in(e  +  <«  =  ^- 

To  express  this  ratio  in  terms  of  0  and  <^,  draw  AC  per- 
pendicular to  the  initial  line  of  </>,  and  CD  perpendicular  to  OX. 
Then  the  functions  of  B  and  <^  may  be  found  from  the  right 
triangles  OT>C  and  OCA  respectively. 

Draw  CE  perpendicular  to  AB.  Then  the  triangles  ODC 
and  AEC  are  similar  (why?),  so  that  Z.EAC  =  6,  and  the 
functions  of  6  may  also  be  found  from  the  triangle  AEC, 

We  then  have 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      339 
BA 


sin  {6  +  0) 


OA 

DC  +  EA 

OA 

PC  sin  e  +  CA  cos  B 
OA 

.    nOC  ^         ^CA 


Fig.  196. 


or  sin  (^  +  </>)  =  sin  ^  cos  0  +  cos  ^  sin  <j>. 

In  like  manner, 


1) 


cos(e  +  4>)-^ 


SO  that 


OP-  EC 
OA 

PC  COS  d  -CA  sin  6 
OA 

nOC         •     nCA 
cos  a  7^-r  -  sin  ^  TT-j 

OA  OA 


cos  (^  +  <^)  =  cos  6  cos  (l>  -  sia.  6  sin  0. 


(2) 


340  ELEMENTARY  FUNCTIONS 

The  proofs  hold  for  either  figure.  In  deriving  (2)  when 
B  -\-  (j)  is  obtuse,  it  should  be  noted  that  OB  is  negative,  so 
that  we  will  still  have  OB  =  OD  -  BD. 

We  shall  assume  that  these  formulas  hold  for  all  values  of 
6  and  </>,  negative  as  well  as  positive.    (See  Exercises  5-8  below.) 

Dividing  (1)  by  (2),  and  using  (4),  page  333,  we  have 

ffi      ^^  -  sJQ  (^  +  </>)  _  sin  6  cos  (f)  +  cos  6  sin  0 
tan  (    +q>)  -  ^^^  (^q  ^  ^^  -  (>os  0  cos  <^  -  sin  6  sin  (j)' 

Dividing  numerator  and  denominator  by  cos  6  cos  </>,  we 

obtain 

sin  6      sin  <^ 

//I  .  j.\         cos  6      cos  (b 

tan  (^  +  <^)  = : — s — ^^' 

1       sm  0    sm  (t> 

cos  6    cos  0 

,  ,       ,^  ,    , .       tan  0  +  tan  0  ,_ . 

whence        tan  id  +  <t>)=^  l^tan^tanV  ^^^ 


EXERCISES 

1.  Express  formulas  (1),  (2),  (3),  in  words. 

2.  Using  the  functions  of  30°,  45**,  60°,  as  given  on  page  161,  find  all 
the  functions  of  (a)  75°,  (b)  105°.     Hint:  sin  75°  =  sin  (45°  +  30°),  etc. 

3.  If  ^  =  arc  sin  f  and  <})  =  arc  cos  /a,  find  the  functions  of  6  +  (f>. 

4.  What  force  acting  parallel  to  the  plane  is  necessary  to  support  a 
body  weighing  100  pounds  on  a  smooth  plane  if  the  inclination  is  ^  =  arc 
sin  f  ?     If  the  inclination  is  <^  =  arc  sin  jf  ?     If  the  incUnation  is  ^  +  ^? 

6.  Prove  that  (1)  holds  for  all  positive  values  of  0  and  <f>. 

Solution:  We  will  first  prove  that:  //  (1)  and  (2)  are  true  for  a  pair 
of  values  6  and  <t>,  then  (1)  is  true  when  6  is  increased  by  90°. 

Let  6'  =  90°  +  d. 

Then  sin  (6'  +  0)  -  sin  (90°  +  0  +  <f>) 

=  cos  (^  +  </)) 

=  cos  6  cos  (j)  —  sin  B  sin  <f> 

=  cos  {9'  -  90°)  cos  (j)  -  sin  {$'  -  90°)  sin  0 

=  cos  (90°  -  6')  cos  </)  +  sin  (90°  -  d')  sin  <f> 

=  sin  0'  cos  (j)  +  cos  6'  sin  </>. 

Since  (1)  holds  for  all  acute  values  of  6  and  <f>,  by  the  above  it  holds 
for  all  obtuse  values  of  0  and  all  acute  values  of  0. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       341 

Applying  the  fact  proved  above  once  more,  (1)  holds  for  all  values  of  d 
less  than  270°  and  of  (f)  less  than  90°;  etc. 

In  Uke  manner  we  may  prove  that  <f>  may  be  increased  by  90°  and 
thence  that  (1)  holds  for  all  positive  values  of  6  and  <t>. 

6.  Prove  that  (2)  holds  for  all  positive  values  of  6  and  (f). 

7.  Prove  that  (1)  holds  if  (f>  is  negative. 

Solution:  No  matter  what  the  value  of  0  may  be,  we  can  choose  an 
integer  n  such  that  n  360°  +  ^  is  positive. 

Then  sin  (0  +  <^)  =  sin  {n  360°  +  6 +  <!>)=  an  Id  +  {n  360°  +  <^)] 

=  sin  d  cos  (n  360°  +  <f))  +  cos  6  sin  (n  360°  +  (f)) 

=  sin  0  cos  ^  +  cos  ^  sin  ^. 

8.  Prove  that  (2)  holds  for  all  negative  values  of  <j}. 

9.  Is  it  necessary  to  employ  the  methods  of  Exercises  5-8  to  prove 
that  (3)  holds  for  all  values  of  6  and  <f>,  positive  and  negative? 

10.  What  'property  of  (a)  a;**,  where  n  is  a  positive  integer;  (b)  6*, 
is  analogous  to  the  properties  of  sin  x,  cos  x,  and  tan  x  given  by  formulas 
(1),  (2),  and  (3)? 

11.  If  sin  1°  is  known,  how  complete  a  table  of  values  of  the  trigo- 
nometric functions  can  be  computed? 

12.  Prove  the  identities: 


(a)  Sin    (^  +  ^)  -  cos  (^  +  ^)  =  sii 

(b)Tan(|+x)  =  ][-±- 


+  tana; 
tanx 


(c)  Sin  {x  +  y)  cos  y  +  coa{x  +  y)  sin  t/  =»  ein  (a;  +  2y). 
13.  Solve  the  equations: 


(a)  Sill  f  a:  +  M  = 

(b)Sin(a:  +  |)-cos(a;  +  ^)  =  ^ 


V2 
2  * 

V2 
2  * 


14.  Prove  that  the  force  to  make  a  sailboat  move  forward  will  be 
greatest  when  the  direction  of  the  sail  bisects  the  angle  between  the  keel 
and  the  apparent  direction  of  the  wind. 

121.  Fimctions  of  the  Difference  of  Two  Angles.    We  now 

seek  an  expression  for  the  sine  of  the  difference  of  two  angles, 
analogous  to  the  formula  in  algebra  for  the  square  of  the  dif- 
ference of  two  numbers. 


342  ELEMENTARY  FUNCTIONS 

Since  6  -  (j)  =  6  +  (-  (jy),  and  since  the  formulas  in  the  pre- 
ceding section  are  true  for  all  values  of  the  angles,  we  have 
sin  (0  -  0)  =  sin  {d  +  (-  0)) 

=  sin  6  cos  (-  0)  +  cos  6  sin  (-  </>). 

But      cos  (-  <^)  =  COS0    and    sin  (-</>)  =  -  sin  0. 
And  hence, 

sin  (6  -<!))=  sin  6  cos  (t>  -  cos  6  sin  0.  (1) 

In  like  manner  it  may  be  proved  that 

cos  {6  -  <j>)  =  cos  0  cos  <^  -f  sin  ^  sin  0,  (2) 

122.  Functions  of  Twice  an  Angle,  or  the  Functions  of  Any 
Angle  in  Terms  of  Half  the  Angle.    Since  20  =  ^  +  0,  we  have 

sin  26  =  sin  {B  +  6) 

=  sin  0  cos  0  +  cos  6  sin  6,  (1),  page  339, 

and  hence 

sin  20  =  2  sin  0  cos  0.  (1) 

In  like  manner,  it  may  be  proved  that 

cos  26  =  cos2  6  -  sin2  0,  (2) 

a.d  tan2»  =  j?^.  (3) 

Since  sin^  6  +  cos^  0  =  1,  (2)  may  be  written  in  the  forms 

cos  20  =  (1  -  sin2  6)  -  sin^  0  =  1  -  2  sin^  0.         (2a) 
cos  20  =  cos2  0  -  (1  -  cos2  0)  =  2  cos^  0-1.        (2b) 

The  graph  of  sin  20  may  be  obtained  by  bisecting  the 
abscissas  of  points  on  the  graph  of  sin  0  (Theorem,  page  151). 
The  graph  suggests  that  the  period  (Definition,  page  168)  of 
sin  20  is  TT  =  180°.  This  is,  indeed,  the  case.  For  if  we  re- 
place 0  by  0  +  180°,  we  get 

sin  2(0  +  180°)  =  sin  (20  +  360°)  =  sm  20. 

A  general  expression  for  the  sine  of  the  product  of  two 
numbers  would  be  analogous  to  the  theorem  giving  the  log- 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       343 


arithm  of  the  product  of  two  numbers  (page  223),  or  the  nth 
power  of  the  product  of  two  numbers  [(3),  page  153].  Formula 
(1)  is  the  special  case  of  such  a  theorem  obtained  when  one 


Fia.  197. 

of  the  numbers  is  2.    Another  special  case  is  given  in  equation 
(1)  of  the  section  following. 

123.  Functions  of  Half  an  Angle,  or  Functions  of  Any  Angle 
in  Terms  of  Functions  of  Twice  the  Angle.  Since  all  that  is 
essential  in  the  formulas  of  the  preceding  section  is  that  the 
angle  on  the  left  be  double  that  on  the  right,  formulas  (2a) 
and  (2b)  may  be  written 

cos  0  =  1  -2sin2  612, 
cos  6  =  2cos2  612-  1. 

Solving  these  equations  for  sin  6  /2  and  cos  6  /2  we  have 


sin  612 


cos  612 


cos  6 


+  cos  6 


Dividing  (1)  by  (2)  and  using  the  formula  (4),  page  333, 


tan 


'i^-4\t 


cos  6 


cos  6 


(1) 


(2) 


(3) 


EXERCISES 

1.  Express  the  formulas  in  Sections  121,  122,  and  123  in  words.  In 
Section  122,  describe  (a)  B  as  any  angle,  (b)  20  as  any  angle.  In  Section 
123,  describe  (a)  6  as  any  angle,  (b)  B/2  as  any  angle. 

2.  Find  all  the  functions  of  15°  from  those  of  45°  and  30°. 


344  ELEMENTARY  FUNCTIONS 

3.  Find  the  functions  of  7r/8  from  those  of  7r/4;  of  7r/12  from  those  of 
7r/6. 

The  table  given  by  the  Hindu  Aryabhata  (476  a.d.  -)  gives  the  values 
of  the  sines  of  angles  at  intervals  of  3°  45'.  How  could  this  table  be 
obtained? 

4.  If  ^  =  90",  show  that  (1),  (2),  (3),  Section  121,  reduce  to  formulas 
in  Section  62,  page  177. 

5.  If  ^  =  180°,  show  that  (1),  (2),  (3),  Section  121,  reduce  to  (1),  (2), 
(3),  page  192. 

6.  State  the  properties  of  7^  analogous  to  each  of  the  equations  (1)  in 
Sections  120,  121,  122,  123. 

7.  In  the  following,  find  sin  B  and  cos  B  without  finding  B  given 

(a)  Tan  2B  =  f.  (b)  tan  2^  =  -  f .  (c)  tan  2B  =  -V- 

Hird:  Find  cos  2B  from  a  figure  (see  Exercise  10,  page  170)  and  then 
use  formulas  (1),  and  (2),  Section  123. 

8.  Express  sin  4^  in  terms  of  functions  of  20;  sin  B  in  terms  of  functions 
of  B/2',   sin  30  in  terms  of  functions  of  30/2. 

9.  Transform  (3),  Section  123,  into  the  forms 

ra  /ON     1  -  cos  B        sin  B 
tan  (0/2) 


sin  0        1  +  cos  0 

Then  transform  each  of  the  fractions  into  tan  (0/2)  by  expressing  sin  6 
and  cos  0  in  terms  of  0/2,  using  the  formulas  in  Section  122.  In  what 
respect  is  the  latter  procedure  preferable? 

10.  If  tan  20  =  1.4123,  find  0  and  cos  0/2. 

11.  A  body  is  placed  on  a  rough  plane  which  is  incUned  at  any  angle 
greater  than  the  angle  of  friction.  (The  angle  of  friction  is  an  angle  0 
such  that  tan  (f)  =  m,  the  coeflficient  of  friction.)  If  the  body  is  supported 
by  a  force  acting  parallel  to  the  plane,  find  the  limits  between  which  the 
force  must  he. 

Let  0  be  the  angle  of  inclination  of  the  plane,  W  the  weight  of  the  body 
and  R  the  reaction  perpendicular  to  the  plane. 

(a)  Let  the  body  be  on  the  point  of  moving  down  the  plane,  so  that  the 
force  of  friction  acts  up  the  plane  and  is  equal  to  mR.  Let  P  be  the  force 
required  to  keep  the  body  at  rest. 

Resolving  W  into  components  parallel  and  perpendicular  to  the  plane, 
we  have  P  +  mR  =  W  sin  B,  R  =  W  cos  0,  and  therefore  P  =  W  (sin  B  -  m 
cos  0).    Since  m  =  tan  0  we  have,  P  «=  PT  (sin  0  -  tan  <f)  cos  0). 

„         p     ^  sin  0  cos  <^  -  sin  <}>  cos  B     ,^  sin  (0  -  <f>) 
cos  <l>  cos  (f> 

(b)  Let  the  body  be  on  the  point  of  motion  up  the  plane.  Complete  the 
solution. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       345 

12.  What  is  the  maximum  and  the  minimum  force  which  will  hold  a 
weight  of  12  pounds  on  a  plane  incUned  at  an  angle  of  40°,  if  the  coefficient 
of  friction  is  0.5,  and  if  the  force  acts  parallel  to  the  plane? 

13.  A  block  W  rests  on  a  horizontal  plane.  If  an  obhqae  force  P  acts 
upon  W,  making  an  angle  6  with  the  direction  of  sUding,  and  if  the  coef- 
ficient of  friction  is  m  =  tan  (f),  prove  that  the  magnitude  of  P  that  will 
cause  the  block  to  shde  is 

sin  (/) 


P  =  W 


cos  {d  -  (t>y 


Find  the  least  pull  that  will  make  the  block  slide. 
Suggestion:  Show  that  P  will  be  a  minimum  when  B  =  <f),  i.e.,  the  direc- 
tion of  pull  is  given  by  the  angle  of  friction. 

14.  If  the  inchnation  of  a  plane  is  ^  =  arc  sin  \,  what  horizontal  force 
would  support  a  body  of  50  pounds  upon  it?  What  horizontal  force  would 
be  required  if  B  were  doubled? 

15.  On  the  same  axes,  plot  the  graphs  of  the  functions  below.  Find 
the  period  of  each  function. 

(a)  cos  B  and  cos  2B.  (f)    sin  B  and  sin  3^. 

(b)  sin   B  and  sin  {B/2).  (g)   cos  B  and  cos  3^. 

(c)  tan  B  and  tan  2B.  (h)  sin  B  and  2  sin  {B/2). 

(d)  cos  B  and  cos  {B/2).  (i)  cos  B  and  3  cos  2B. 

(e)  tan  B  and  tan  {B/2).  (j)  tan  B  and  \  tan  {B/2). 

16.  Solve  the  equations: 

(a)  sin  2x  +  cos  x  =  0.  (d)  sin  2a:  -  2  sin  a;  =  0. 

(b)  tan  2a;  +  tan  a;  =  0.  (e)  cos  2a;  +  sin  x  =  0. 

(c)  2  tan2  a;  -  3  tan  a;  +  1  =  0.  (f)    cot  2a;  +  tan  a;  =  0. 

17.  Prove  the  identities: 

(a)  (sin  a;  +  cos  a;)2  =  1  +  sin  2a;. 
,,  .       2  tan  X         .    ^ 

.  .         sin  2^  ^.       a 

(c)  :; ~Q  =  tan  d. 

^  ^     1  +  cos  2^ 

(d)  cos  {x  +  y)  cos  {x  -y)  =  cos^  x  -  sin^  y. 

(e)  sin  a  cos  (jS  -  a)  +  cos  a  sin  (|8  -  a)  =  sin  /?. 

124.  Slim  and  Difference  of  the  Sines  or  Cosines  of  Two 
Angles.  To  express  the  sum  of  the  sines  of  two  angles,  sin  a 
+  sin  j3,  as  a  product,  let 

a  =  e  +  <i> 
and  p  =  6  -(t). 


346  ELEMENTARY  FUNCTIONS 

Then       sin  a  +  sin  /3  =  sin(  6  +  (j))  +  sin  {d  -  0) 
=  sin  6  cos  4>  +  cos  ^  sin  <^ 

+  sin  6  cos  <^  -  cos  ^  sin  <^ 
=  2  sin  ^  cos  (j). 

Solving  the  first  two  equations  for  6  and  </>,  by  adding  and 
subtracting,  and  dividing  by  2,  we  have 

e  =  i{a  +  ^),  0  =  i(a  -  ^), 

and  hence,  substituting, 

sin  a  +  sin  /3  =  2  sin  i(a  +  1^)  cos  J(a  -  jS).  (1) 

In  hke  manner,  prove 

sin  a  -  sin  jS  =  2  cos  J(a  +  jS)  sin  |(a  -  /3).  (2) 

cos  a  +  cos  jS  =  2  cos  i(a  +  ^3)  cos  i(a  -  ^),  (3) 

cos  a  -  cos  jS  =  -  2  sin  ^{a  +  /3)  sin  i(a  -  ^3) .  (4) 

Equations  (1)  and  (2)  give  properties  of  the  function  sin  x. 
What  are  the  analogous  properties  of  log  x?  To  what  property 
of  x^  is  (2)  analogous? 

125.  Logarithmic  Solution  of  Triangles,  Case  III.  The  law 
of  sines  (page  201)  may  be  written 

sin  A  _  a 
sin  B      b 

,  Applying  division  and  composition  to  this  proportion 

sin  A  —  sin  jB  _a  —  h 

sin  A  +  sin  B  ~  a  +  6 

Then  by  (2)  and  (1),  Section  124, 

2  cos  \{A  +  B)  sin  \{A  -  B)      a-h 


2  sin  J(A  +  B)  cos  \{A  -  B)      a  +  6 
From  (6),  (1),  and  (4),  page  333,  we  then  have 

tan  \{A  -  B)  _  a-h 
tan  i(A  +  j5)  ~  a  +  6 

whence        tan  |(il --5)  =  ^-^  tan  |  (il +5).  (1) 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      347 

If  two  sides  and  the  included  angle  of  a  triangle  are  given, 
the  other  angles  may  be  found  by  means  of  (1),  as  is  the 


Example.    Solve  the  triangle,  given 

6-    96,         ^- 

.  34°.24. 

We  have:                   a  +  6  =»  214, 
a-h=    22, 

Then  by  (1), 

-  72°.88. 

tan  \{A  -B)=^  tan  72^88. 

log  22  =  1.3424 
log  tan  72^88  =  0.5115 

2.8539  -  1 
log  214  =  2.3304 

p^ 

log  tan  K^  -B)  =0.5235-1. 
Hence  h{A  -  B)  =  18°.46. 
But      KA+B)=72^88, 

6  = 

FlQ, 

=  96 

.  198. 

and  hence           A  =  91^34  (adding) 
and                      B  -  54^42  (subtracting) 

To  find  c,  we  have 

6  sin  e      96  X  sm  34°.24 
^       sin  i^          sin  54^42 
log  96  =  1.9823 

log  sin  34''.24  =  0.7502  -  1 
2.7325  -  1 

log  sin  54^42  =  0.9102  -  1 
log  c    =  1.8223 
whence                     c  =  66.42. 

Check.    Find  one  of  the  given  sides  from 

c,  A  or  B,  and  C. 

c  sin  A      66.42  x  sin  91*'.34 
sin  C              sin  54°.42 

log  66.42  =  1.8223 
log  sin  9r.34  =  0.9999  -  1 
2.8222  -  1 
log  sin  34°.24  =  0.7502  -  1 
log  a  =  2.0720 
a  =  118.0, 

which  agrees  with  the  given  value  of  a. 


348  ELEMENTARY  FUNCTIONS 

126.  Logarithmic  Solution  of  Triangles,  Case  IV.  We  seek 
an  expression  for  tan  ^A,  A  being  an  angle  of  a  triangle  ABC, 
in  terms  of  the  sides.    We  have 


/I  —  cos  A 
VlT^^  (by  (3),  page  343), 


I         b^  +  <^  -a 


26c  (substituting  the  value  of  cos  A 

en  by  the  law  of  cosines) 

(by   multiplying    numerator 


6M-c2--a2  given  by  the  law  of  cosines) 


=  V(6^  +  26c  +  o^)-a^   ^^^  denominator  by  26c,  and 
^  groupmg  the  terms) 

-4 


(a-6+c)(a  +  6-c)     (f^^^^rfng) 
(a  +  6  +  c)  (6  +  c  -  a)  ^^ 

Denote  half  the  perimeter  by  s,  so  that 

a  +  6  +  c  =  2s. 
Subtracting  26,  a  -  6  +  c  =  2s  -  26  =  2(s  -  6). 

Subtracting  2c,  a  +  6  -  c  =  2(s  -  c). 

Subtracting  2a,  6  +  c  —  a  =  2(s  —  a). 

Substituting  above, 


^.A=,^^^^^^ 


i 


2s2(s  -  a) 


-4 


(s  -  a)(s  -  6)(s  -  c) 
s{s  -  ay 


7^a\ 


(s  -  a)(s  -  6)(s  -  c) 


Hence 


tan^il  = f         where      r  =  y- — — (1; 

s  -  a  V  s  ^  ^ 

Interchanging  the  values  of  a,  6,  c,  does  not  affect  the  values 
of  8  and  r  and  hence 

tan  J5 ^ ,    tan  K  =  -^ —  (2) 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      349 

It  is  proved  in  most  elementary  geometries  that  the  area  of 
a  triangle  in  terms  of  its  sides  is 

A  =  Vs{s  -  a){s  -  b){s  -  c), 
whence  A  =  rs,  (3) 

A  simpler  proof  is  indicated  in  Exercises  6  and  7  below. 

Example.    Find  the  angles  and  area  of  the  triangle  a  =  34.25,  b  =  42.91, 
c  =  50.82. 

The  angles  are  found  by  (1)  and  (2),  the  first  step  in  the  computation 
being  to  find  log  r  from  its  value  in  (1). 


a=    34.25 

log  (s  -a)  =  1.4734 

6=    42.91 

log(s-6)  =1.3239 

c=    50.82 

log(s-c)  =1.1196 

2s  =  127.98 

3.9169 

s=    63.99 

log  8  =  1.8061 

s-a=    29.74 

2  [2.1108 

s-6=    21.08 

logr  =  1.0554 

s-c=    13.17 

logr            =1.0554 

logr            =1.0554 

logr             =1.0554 

log  {s-a)  =  1.4734 

log(s-6)   =  1.3239 

log  (s-c)    =  1.1196 

log  tan  M  =  0.5820-1 

log  tan  1  B  =  0.7315  - 

-1   log  tan  ^  C  =  0.9358  -  1 

|A  =  20^90, 

iB  =  28°.32 

^C  =  40°.78. 

A  =  41°.80, 

B  =  56^64 

C  =  8r.56. 

Check.    A+B  +  C  = 

180^00. 

By  (3),  the  area  is  A 

=  rs 

log  r  =  1.0554 

=  727.0. 

log  s   =  1.8061 
log  A  =  2.8616 

EXERCISES 

1.  Solve  the  equations: 

(a)  sin  3a;  +  sin  a:  =  0.    Hint:  Use  (1),  page  346. 

(b)  cos  3a;  +  cos  x  =  0. 

2.  Prove  the  identities: 

(a)  sin  f  ^  +  a;  j  -  sin  f  -  -  a;  j  =  sm  x. 

...    sin  3a;  +  sin  a;       ,      „ 

(b)  5 — ■ =  tan  2a;. 

cos  3a;  +  cos  x 

3.  Solve  and  check  the  following  triangles,  and  find  the  areas. 

(a)  a  =  34.34,  6  =  18.96,  C  =  50°.68. 

(b)  a  =  543. 2,  6  =  496. 7,  c  =  568. 3, 

(c)  b  =  7634,  c  =  8427,      A  =  102°.  16. 

(d)  a  =  1243,  b  -  1497,  -   c  =  2046. 


350 


ELEMENTARY  FUNCTIONS 


4.  Two  men  start  to  walk  from  the  same  point  in  directions  inclined  at 
45°  to  each  other.  When  one  has  gone  792  yards,  the  other  has  gone  846 
yards.  What  is  the  inclination  of  the  line  joining  them  to  their  paths, 
and  how  far  apart  are  they? 

5.  The  sides  of  a  triangular  field  are  84.3  rods,  77.5  rods,  and  102.1 
rods.  What  are  the  angles  of  the  field?  How  many  potatoes  could  be 
raised  on  it,  if  the  yield  were  200  bushels  per  acre? 

6.  Find  the  radius  of  the  circle 
inscribed  in  a  triangle  in  terms 
of  its  sides. 

Let  the  points  of  contact  with 
the  sides  be  L,  M,  N,    Then 

AN  =  AM,  BL  =  BN,  CL  =  CM, 

hence 

s^AN  +  BL  +  CL^-AN  +  a, 
whence 

AN  =  s  -  a. 

Then  if  r'  is  the  radius  and  0 
the  center  of  the  circle,  we  have 


tan  2^  A  = 


AN 


But  tan 


U' 


r 

» 

-a 


and  hence 


■/ 


s  -a)  (s  -  6)  (s  -  c) 


7.  Prove  that  the  area  of  ABC  is  A  =  rs.  Hint:  If  0  is  the  center  of 
the  inscribed  circle,  ABC  =  OAB  +  OBC  +  OCA. 

8.  Find  the  diameter  of  the  circle  circumscribed  about  a  triangle. 
Hint:  Draw  the  diameter  CD  through  C,  and  join  D  to  B.  What  are  the 
values  of  the  angles  CBD  and  CDB1 

9.  A  and  B  are  two  inaccessible 
points.  If  CD  =  1000  yards,  ZACD 
=  82^34,  ZADC  =  58^22,  ZBCD 
=  72°.78,  ZBDC  =  105M3,  find  the 
direction  of  AB  (the  angle  AB  makes 
with  AC)  and  the  distance  from  A  to  B. 

10.  The  altitude  of  a  triangular 
prism  is  13.48  inches,  and  the  sides  of 
the  base  are  11.23  inches,  8.43  inches, 
and  4.79  inches.    Find  the  volume. 

11.  The  sides  of  a  field  measured  in 
succession    are   22.5,  20.4,    18.7,  and 

21.5  rods.    The  angle  between  the  first  two  sides  is  84°.5.    How  many 
acres  does  the  field  contain? 


Fig.  200. 


I 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      351 
127.  Miscellaneous  Identities  and  Equations. 

Example  1.  Express  sin  3^  in  terms  of  sin  6.  Since  SO  =  20  +  8,  we 
have 

sin  30  =  sin  (2^  +  ^)  =  sin  26  cos  0  +  cos  20  sin  0 

=  (2  sin  0  cos  0)  cos  0  +  (cos^  0  -  sin^  0)  sin  0 

=  3  sin  0  cos2  0  -  sin3  0 

=  3  sin  0  (1  -  sin2  0)  -  sin'  0 

=  3  sin  0  -  4  sin'  0. 

This  identity,  like  the  formula  for  sin  2^,  is  a  special  case  of 
the  sine  of  the  product  of  two  nmnbers  (compare  (3),  page  153, 
and  (7),  page  223.  It  may  be  used  in  the  proof  of  the  fact 
that  it  is  impossible  to  trisect  an  angle  with  ruler  and  compasses, 
as  this  phrase  was  understood  by  the  ancient  Greeks. 

^  o     T»         xu    -J     x-x    tan  2x  +  tan  x     sin  Sx 

Example  2.    Prove  the  identity  j^ — ^ =  — : • 

tan  2x  -  tan  x      sin  x 

sin  2x     sin  x 

^     .  tan  2x  +  tan  x      cos  2x     cos  x 

Proof.  7 — n 1 z  =  "■ — n = 

tan  2x  -  tan  x     sin  2x      sin  x 

cos  2x      cos  X 

_  sin  2x  cos  x  +  cos  2x  sin  x 

~  sin  2x  cos  x  -  cos  2x  sin  x 

(by  multiplying  numerator  and  denominator  by  cos  2x  cos  x) 

sin  (2x  +  x)  _  sin  Sx 
'^  sin  (2x  -  x)  ~  sin  X 

Example  3.     Solve  the  equation  cos  x  —  cos  2x  +  cos  3x  =  0. 

Applying  the  formula  for  the  sum  of  the  cosines  of  two  angles  to  the  first 
and  third  terms,  we  get  ' 

2  cos  2x  cos  X  -  cos  2x  =  0. 

Factoring,  cos  2x  (2  cos  x  -  1)     =0, 

whence  cos  2a:  =  0  or     cos  a;  =  |. 

TT  TT 

Then  2a;  =  ±  -  +  2mr        or  a:  =  =*=  -5  +  2n7r. 

Hence  x  =  ^  -:  +mr  or  a;==*=^  +  2n7r. 

4  3 

EXERCISES 

1.  Prove  the  following  indentities. 

/  X    ^  X  sin  (x  +  y)  /un       i.      .      x         sin  (a;  +  y) 

(a)    tan  a;  +  tan  2/  = ^  •         (b)    cot  x  +  coty  =   .    ^   ^ -'^^ . 

^  '  'J/      gQg  ^  gQg  ^  Sin  a:  Sin  iy 


352  ELEMENTARY  FUNCTIONS 

.  ,        sina;  +  sin  2x         .  T'  sin  x  ^x 

(C)      1 ; ^  =  tan  X.  (k) =  cot  p:  • 

^       1  +  cos  a;  +  cos  2a;  1  -  cos  a;  2 

,,.     sin  x  +  sin  2a;  +  sin  3a;      ,       -  ,  .       .        .    _ 

(d)    ; ^ — ■ 5-  =  tan  2x.   /,>.         +  /     ,  ttX      1  -  sin  2a; 

cos  X  +  cos  2x  +  cos  3a;  (1)      cot  I  a;  +  -  I  = 

\        2/         cos  2a; 

sin  A  +  sin  ^  _         1  .  . 
^®^    cos  A  +  cos  5  ~  *^°  2  ^"^  "^  ^''-     (m)  cot  a;  +  tan  a;  =  2  esc  2a;. 


(f)  1  +  tan  2a;  tan  x  =  sec  2a;. 

.X.         X 

cot  ^  -  tan  .J 

(g)    r Z  =  cos  a; 


n)    sin  f  ^  +  a;  j  =  cos  f  ^  -  a;  y 


X             X  =  ^^°  •^-                       /  ^  +       /^  .  7r\         VI  +sin  a; 

,,  >        sin  2a;         ,  3. 

W    ; FT  =  tan  a;.  o  +„„  _ 

^  ^    1  +  cos  2a;  ^  xan  ^                  , 

(p)    =  sin  X. 


(i)     tan  X 


2  cot  I  1  +  taii2 1 


cot2  -  -  1  ,.    tan  3a;  -  tan  X  1 

(q) 


2 

X 


tan  3a;  +  tan  x     2  cos  2a; 
.      tan^l 
(j)     f  =  cog  a; .  (r)    cos  3a;  =  4  cos'  x  -  3  cos  x. 

1  +  tan2 1 

(s)    2  sin  f  X  +  2  )  sin  (  2;  -  7  J  =  sin^  x  -  cos^  x. 

2.  Solve  the  following  equations. 

(a)  cos  2x  -  2  cos  X  -  3  =  0.  (h)  sin  2x  +  sin  3x  =  0. 

(b)  cos  2x  -  cos  X  =  0.  (i)  sin  3x  +  sin  5x  =  0. 

(c)  cos  2x  -  sin  X  =  0.  (j)  2  sin  x  -  3  cos  x  =  1. 

(d)  tan  2x  =  s  in  x  sec  x.  (k)  sin  x  +  cos  x  =  \/2. 

(e)  12  sec2  x  +  7  tan  x  -  24  =  0.  (1)  4  sin  x  +  3  cos  x  =  5. 

(f )  sin  X  +  sin  2x  +  sin  3x  =  0.  (m)  tan  x  +  tan  2x  +  tan  3x  =«  0. 

(g)  cos  X  4-  sin  2x  -  cos  3x  -  0.       ,  .         ,  x       .  ,  x      ,      „       , 

(n)    cos2  2  -  sin^  2  =  1  -  2  cos^  x. 

3.  In  the  following  identities  transform  the  left  member  into  the  right. 

(a)  sin2  X  =  i  -  ^  cos  2x.  (b)  cos^  x  =  i  +  |  cos  2x. 

(c)  sin*  X  =  f  -  I  cos  2x  +  I  cos  4x.  (d)  cos*  x  =  f  4-  ^  cos  2x  + 1  cos  4x. 

128.  Differentiation  of  Trigonometric  Functions.    In  order 
to  find  the  derivative  of  sin  x  we  shall  need  to  show  that 

lim     !!]L^_i  ■     m 

0^0       e     ~  ^^" 


I 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       353 

For  this  purpose  we  construct  the  figure,  which  is  self-ex- 
planatory, and  assume  that  P'P<  arc  P'  NP<Q'Q. 
Dividing  by  2, 

MP<arc  NP<NQ. 

Substituting 

MP  =  OP  sin  0  =  r  sin  6, 
NQ  =  ON  tan  ^  =  r  tan  6, 
and 

arc  NP  =  rO  (Theorem,  page  171), 

we  get 

r  sin  d<rd<r  tan  6, 

Dividing  by  r  sin  6, 

i<J-.<  ' 


Fig.  201. 


sin  6     cos  d 

As  6  approaches  zero,  cos  6  approaches  1,  and  so  also  does 
1  /cos  6.  Hence  6  /sin  6,  which  Hes  between  1  and  1  /cos  dj 
approaches  1,  and  hence  the  reciprocal  function,  sin  6/6, 
approaches  1. 

Theorem  1.  If  u  is  a  function  of  x,  then  the  derivative  of  sin  u 
with  respect  to  x  is  cos  u  times  the  derivative  of  u  with  respect 
to  X,  that  is, 

Dxsmu  =  cos  u  DxU,  •  (2) 

Let  y  =  sin  u. 

Then  y  -\-  Ay  =  sin  {u  4-  Au). 

Subtracting,         Ay  =  sin  {u  +  Au)  —  sin  u 


=  2  cos 


(»-*) 


^"^  sin  ^  [(2),  page  346.] 


Au 
Multiplying  and  dividing  by  -x- ' 


.    Au 
sm  -rr 


Ay  =  COS  (u  +  ■YJ-ST'  '^"- 


354  ELEMENTARY  FUNCTIONS 

Dividing  by  ^x,  ^  =  cos  [u  +  -^j  —^ 


In  passing  to  the  limit  as  Ax  approaches  zero,  we  notice  that 

Aw  approaches  zero,  and  so  also  does  — ^.     Then  the  limit  of 

the  second  factor  on  the  right  is  unity,  by  (1)  above.     Passing 
to  the  limit,  we  get 

DxV  =  cos  u  DxU. 

Corollary  1.     //     y  =  sin  x,  then  Dxy  =  cos  x,  (3) 

Corollary  2.     //  Dxy  =  cos  x,  then  y  =  sin  x  +  C,  (4) 

and  if                     Dxy  =  cos  u  DxU,  then  y  =  sin  u  +  C.  (5) 

Theorem  2.    Dxcos  u  =  -  sin  i/  DxU.  (6) 

By  (1),  page  177,  we  have  ] 

2/  =  cos  w  =  sin  ( -  -  uj, 
and  hence  Da,y  =  D^  sin  [^  -  uj 

=  cos(|-t.)D.(|-u)  by  (2) 

=  —  sin  w  DxU. 

Corollary  1.    //      y  =  cos  x,  then  Dxy  =  -  sin  jc.  (7) 

Corollary  2.    If  Dxy  =  sin  x,  then  y  =  -  cos  x  +  C,  (8) 

and  if  Dxy  =  sin  u  DxUy  then  y  =  —  cos  u  -\-  C.  (9) 

Example  1.     Uniform  motion  in  a  circle.     If  the  position  of  a  point 
P{x,  y)  at  the  time  t,  is  given  by  the  equations 

X  =>  a  cos  (at,  y  =  a  sin  Oit, 

show  that  P  moves  in  a  circle,  and  find  the  magnitude  and  direction  of 
the  velocity  and  acceleration. 

Squaring  and  adding  the  given  equations,  we  get 

x^  +  y^  =  a^  cos*  cot  +  a^  sin*  (at  -  a*, 

which  is  the  equation  of  a  circle  (page  327),  and  the  point  P  therefore 
moves  on  a  circle. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      355 


The  radius  OP  makes  an  angle  oi  6  =  03t  with  the  ar-axis.     Since  Dfi  =  co, 
the  rate  of  change  of  6,  or  the  angular  velocity  of  OP,  is  the  constant  co. 

The  table  of  values  gives  the  times  at 
which  P  crosses  the  axes  in  its  first 
revolution. 

Velocity  of  P.  The  components  of  the 
velocity  parallel  to  the  axes  are  (page  311) 

Vx  =  Dtpc  =  Dt  (a  cos  oit) 

=  -  a  sin  (tit  Dt03t  =  -  aco  sin  co<, 
and 

Vy  =  ao)  cos  cat. 
Then  the  magnitude  of  the  velocity  is 


Fig.  202. 


t?  =  v(-  aco  sin  coO^  +  a^ca^  cos^  o)t  =  aco. 

As  ao)  is  constant,  the  point  P  moves  uniformly  in  its  path.     This  re- 
sult may  be  verified  as  follows:    Since  the  angular  velocity  is  co,  in  one 

second  the  radius  OP  will  sweep  over 
CO  radians,  and  hence  the  arc  described 
by  P  will  be  aoi  (Theorem,  page  171); 
that  is,  P  will  move  a  distance  aco  in 
one  second. 

The  slope  of  the  direction  of  the 
velocity  is 


t 

e=u)t 

X 

y 

0 

0 

a 

0 

TT 

2« 

2 

0 

a 

TT 

IT 

—a 

0 

^ 

3«- 
— 

0 

—a 

27r 
<a 

2^ 

a 

0 

ao)  cos  cot 
-  aco  sin  o)t 


a  cos  0)t 
aeJn  iat 


1 


As  this  is  the  negative  re- 
ciproca-l  of  the  slope  of  OP, 
the  direction  of  the  velocity 
is  perpendicular  to  OP,  and 
hence    it    is    along  the    tangent  Fig.  203. 

Une. 

Acceleration  of  P.     The  components  of  the  acceleration  are 

Qx  =  DtVx  =  —  aco^  cos  (ji)t, 
and  Cy  =  DtVy  =  -  aco^  sin  co<. 


356  ELEMENTARY  FUNCTIONS 

Then  the  magnitude  of  the  acceleration  is 

a  =  Va^oj*  cos^  o)t  +  a^co^  sin^  cot  =  acc^. 
The  slope  of  the  direction  of  the  acceleration  is 


aco^  sin  cot      a  sin  o)t 


—  aof  cos  ixit 


a  cos  oit     X 

As  this  is  the  slope  of  OP,  the  accelera- 
tion acts  along  the  radius.  It  is  directed 
toward  the  center. 

Example  2.  Find  the  area  of  one  arch 
of  the  graph  of  sin  2x  (page  343) . 

From  page  305  we  have 

DxA  =  2/  =  sin  2x. 

As  2a;  is  a  function,  u,  of  x  whose  deriva- 
tive is  2,  we  multiply  and  divide  sin  2x  by 


Fig.  204. 

2,  obtaining 

D,A 

=  i  sin  2x  X  2  =  ^  sin  2x  D^{2x) 

Integrating  by  (9), 

A  =  -  ^  cos  2x  +  C. 

As  A  =  0  when  x  =  0, 

0=  - 

-  5  +  C,        whence        C  =  ^. 

Hence 

A  =  -\cos2x  +  \. 

Substituting  2;  =  ^j  t^®  desired  area  is 

A  =  -  ^  cos  TT  +  I  =  -  K-  1)  +  5  =  1- 
In  finding  the  derivative  of  sin  u,  use  was  made  of  the  fact 

1,  and  in  finding  this  limit  the  angle  6  was 


sin  6 
6 


that  lim 

measured  in  radians.     Hence, 

In  applying  the  rules  for  differentiating  and  integrating  trigo- 
nometric functions  it  must  he  remembered  that  the  independent 
variable  is  measured  in  radians. 


EXERCISES 
1,  Differentiate  the  functions: 


(a)  2  sin  (3x  +  4). 
(d)  3  sin  (27ra;  -  3). 
(g)  sin  (x«). 


(b)  3  cos  4a;. 

(e)  a  sin  {hx  +  c). 

(h)  sin  (x*  +  3x). 


(c)    2  COB  (1  -  TTX). 

(f)   c  cos  {mx  4-  r). 
(i)  2  cos  (a;»  -  x*). 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS       357 

2.  Find  the  derivatives  of  the  following  functions,  by  first  expressing 
them  in  terms  of  the  sine  and  cosine. 

(a)  Dx  tan  u  =  sec*  u  DxU.  (b)  Dx  cot  w  =  —  esc'  u  DxU. 

(c)  Dx  sec  u  =  tan  u  sec  u  DxU.  (d)  Dx  esc  it  =  —  cot  u  esc  u  DxU. 

3.  Find  the  derivative  of  each  of  the  functions  below,  using  the  funda- 
mental method  of  replacing  x  by  x  +  Ax,  etc. 

(a)  cos  X.         (b)  tan  x.  (c)  cot  x.  (d)  sec  x.         (e)  esc  x. 

4.  Find  the  maximum  and  minimum  values  of  each  of  the  trigonomet- 
ric functions  by  the  method  of  differentiation. 

5.  Differentiate  the  following  functions,  using  (5)  page  270. 

(a)  sin*  x.        (b)  cos*  2x.       (c)  sec*  x.  (d)  2  esc*  3x. 

6.  Find  the  coordinates  of  the  points  of  inflection  of  the  graph  of 
(a)  sin  X.  (b)  cos  x.  (c)  tan  x.  (d)  cot  x. 

7.  A  point  moves  so  that  x  =  10  cos  27r<  and  y  =  10  sin  2Trt.  Show  that 
the  point  moves  in  a  circle,  making  one  revolution  per  second.  Find  the 
components  parallel  to  the  axes  and  the  magnitude  and  the  direction  of 
the  velocity  and  acceleration.     Where  does  the  point  start? 

8.  As  in  the  preceding  exercise,  discuss  the  motion  of  a  point  if  x  =  10 
cos  (27r^  +  7r/2)  and  ?/  =  10  sin  {2Trt  +  7r/2).     Where  does  the  point  start? 

9.  Discuss  the  motion  of  a  point  if  x  =  4  cos  (xx/2  +  tt/S),  i/  =  4  sin 
(7rx/2  +  7r/3). 

10.  The  position  of  a  point  P  moving  on  a  circle  of  radius  5  so  that 
OP  rotates  through  2  radians  per  second  is  given  by 

X  =  5  cos  2t,  y  =  5  sin  2t. 

If  M  is  the  projection  of  P  on  the  x-axis  (see  figure  for  Example  1  of 
the  preceding  section),  find  the  position  (s  =  OM),  velocity,  and  accelera- 
tion of  M  at  any  time.  Find  when  and  where  the  values  of  s,  v,  and  a 
are  greatest  and  least. 

11.  A  chip  on  the  surface  of  a  pond  moves  up  and  down  with  the  waves 
according  to  the  law  s  =  sin  ^t.  Find  the  velocity  and  acceleration.  De- 
termine the  maximum  and  minimum  values  of  s,  v,  and  a.  Plot  the  three 
graphs  on  the  same  axes. 

12.  Find  the  area  of  an  arch  of  each  of  the  curves,  (a)  sin  x.  (b)  sin  ^x. 
(c)  cos  2x.     (d)  3  sin  x. 

13.  In  a  right  triangle  ABC,  it  was  found  that  6  =  10  inches,  and 
A  =  45°.     Find  a  and  c,  and  the  error  in  each  due  to  an  error  of  0°.l  in  A. 

14.  In  a  triangle  a  =  2,  6  =  4,  C  =  60°.  Find  c  by  the  law  of  cosines 
and  the  error  in  c  due  to  an  error  of  0°.2  in  C. 

15.  Given  sin  30°  =  ^  =  .5000  and  cos  30°  =  Vd/2  =  .8660,  find  ap- 
proximate values  of  sin  30°.  1  and  cos  30°.  1. 


358 


ELEMENTARY  FUNCTIONS 


16.  Given  sin  45°  =  cos  45"  =  \/2/2  =  .7071,  find  approximate  values  of 
sin  45".!  and  cos  45°.  1. 

17.  Given  sin  60°  =  V3/2  =  .866  and  cos  60°  =  |  =  .500,  compute  a 
three-place  table  of  sines  and  cosines  of  60°.l,  60°.2,  60°.3,  60°.4,  60°.5. 

18.  A  ship  is  anchored  300  yards  from  a  straight  shore,  along  which 
a  searchlight  is  played.  If  the  light  is  turned  imiformly  at  the  rate  of 
one  radian  per  minute,  how  fast  will  the  beam  of  light  be  moving  along  the 
shore  when  it  makes  an  angle  of  30°  with  the  shore? 

19.  A  balloon  is  ascending  vertically  at  the  rate  of  5  feet  per  second. 
An  observer  stands  200  feet  from  the  point  at  which  the  balloon  started. 
How  fast  is  the  angle  of  elevation  changing  when  it  equals  60°? 

20.  A  level  road  approaches  a  hill  300  feet  high.  An  observer  on  the 
hill  notes  that  the  angle  of  depression  of  a  motorcycle  is  80°  and  that  the 
angle  decreases  1°  in  three  seconds.  Find  approximately  the  speed  of 
the  motorcycle. 

129.  Graph    of    the    Function    a  sin  {bx  +  c).    Harmonic 

Curves.    Suppose  at  first  that  c  =  0,  and  consider  the  special 

case 

2/  =  a  sin  hx.  (1) 

The  graph  of  this  function  may  be  obtained  from  that  of 
sin  z  by  first  dividing  the  abscissas  by  b,  which  gives  the  graph 


"P                     V' 

±                  .2        t 

11              Z\              J^ 

-.4                 ^^                 Z 

-1  A        i  c        t  i 

2few    ±          -lUA    ,.    JL.l 

^'^dt            tl^X-        'tU    A 

ignsir         ;  -V        i^-  i: 

it    t         ^^  ^        t     X 

=i--t-  A      ^----^  ^i^     -C  . 

-'^i-     T      1       :    •     \      7^      *     ^ 

JC^  t     ~^-7l         \     1 

-K   ut          2^v.          3^ 

^^ui'-^^-l^yt         ^^] 

\    I    ^^^  \     f               / 

\  i       -  i        H- 

t          t         t 

w'            ti-.z         \i 

Graph  op  a  sin  hx,  a  =  2,  6 
FiQ.  205. 


3. 


of  sin  hx  [(4),  page  152],  and  then  multiplying  the  ordinates 
of  points  on  the  graph  of  sin  hx  by  a  [(3),  page  152].  Since 
the  graph  of  sin  x  repeats  itself  in  intervals  of  27r,  the  graph 
of  (1)  will  repeat  in  intervals  of  27r/6,  so  that  the  function 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      359 

a  sin  bx  is  periodic  with  the  period  27r  /b.  And  since  the  maxi- 
mum and  minimum  values  of  sin  x  are  1  and  —  1,  the  maxir- 
mum  and  minimum  values  of  a  sin  bx  are  a  and  —  a.  The 
constant  a  is  called  the  amplitude  of  the  function. 

The  form  of  the  graph  of  (1)  is  therefore  much  like  that  of 
sin  X,  but  it  differs  from  it  in  period  and  amplitude,  or  in  width 
and  height.  The  graph  may  be  sketched  expeditiously, 
without  drawing  the  graphs  of  sin  x  and  sin  bx,  as  follows: 

Lay  off  OA  (Fig.  205)  on  the  x-axis  equal  to  the  period  27r  /6, 
and  divide  OA  into  four  equal  parts  by  B,  C,  and  D.  At  B  erect 
BE  perpendicular  to  the  x-axis  and  equal  to  a,  and  at  D  draw 
the  ordinate  DF  =  -  a.  Then  the  graph  cuts  the  x-axis  at  0, 
C,  and  A,  has  a  maximum  point  at  E,  and  a  minimum  point 
at  F.  The  part  of  the  curve  from  0  to  A  can  be  drawn  by 
using  these  five  points,  and  the  rest  of  the  curve  may  be  ob- 
tained by  the  periodicity. 

Now  consider  the  general  equation 

2/  =  a  sin  {bx  +  c).  (2) 

It  may  be  written  in  the  form   ,.-^^^Sl  ir?^ 

^-  y  =  asmb(x  +  |j-  (3) 

If  we  set  X  =  a;'  -  7,  which  moves   the  2/-axis   to   the  new 

c 

origin  0'(-  ^,  0),  equation  (3)  becomes 

2/  =  a  sin  bx\  (4) 

Comparing  (4)  with  (1),  we  see  that  the  graph  of  (4)  may 
be  drawn  on  the  new  axes  by  the  method  given  above.  The 
constant  00'  =  -  c/bis  called  the  phase  of  the  function. 

The  graph  of  an  equation  in  the  form  (2)  is  called  a  simple 
harmonic  curve,  and  th-  function  a  simple  harmonic  function. 

Example.     Construct  the  graph  of  2  sin  (  J  a;  +  tt  )• 

Here  a  =  2,  &  =  7r/2,  and  c  =  ir.  Since  the  phase  is  -  c/fe  =  -  2,  we  lay 
ofif  00'  =  -  2  on  the  x-axis,  and  choose  0'  as  a  new  origin.    The  period 


360 


ELEMENTARY  FUNCTIONS 


is  27r/7)  =  4,  and  hence  we  lay  off  O'A  =  4,  and  divide  it  into  four  equal 
parts  by  B,  Ct  D.     Since  the  amplitude  is  o  =  2,  we  erect  the  ordinates 

BE  =  2.  and  DF  =  -  2.  The  graph 
is  then  drawn  through  0',  E,  C,  F,  and 
B.  It  repeats  itself  every  4  units 
along  the  a>-axis. 


I 


t 


:^ 


^ 


? 


A  compound  harmonic  curve  is 
obtained  by  adding  the  ordinates 
of  points  on  two  harmonic  curves. 
It  is  the  graph  of  an  equation  of 
the  form 

2/  =  a  sin  (hx  +  c) 
Fig.  206  +dsm  (ex+f),  (5) 

Fig.  207  shows  the  compound  harmonic  curve 

2  sin  a;  +  sin  (2x  +  tt). 

Harmonic  curves  find  important  apphcations  in  such  theories 
as  sound  and  electricity  which  depend  on  wave  theory.    The 


»                  L 

Z^^ 

-/      % 

2                       ^t          \- 

''       V          \2sBi:-s'n  [2a;  t-7r) 

/      /           *>       \l 

2sinx'        '^               \    \ 

-1       Z    2            „^V 

J-                    ^53                           Z     % 

~   t  I       2       Si              ■       ^^ 

^      ^              ^                  it                          ^                  \ 

tJ              t                 JX^                    -<--          ^^-^ 

O^''                          2               3>l            .           /               /    6  /27f 

..      .  ^     .                 _^  .      .        .                 n.                     .   ^.                 <-       y^ 

''8n2lr-V)            \            /              /       ;       .. 

-1       :^-^                        ifc-^          7       ^ 

J  >:       z^  2 

^^              -4      V 

t    ^     z^  ^ 

-2                                                                    ^            ^>-'^ 

^     t 

^^7 

-.-± 

Fig.  207. 


period  is  frequently  called  the  wave  length.  They  are  applied, 
also,  in  many  motions,  for  example,  in  the  theory  of  th(? 
pendulimi. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      361 

130.  Empirical  Data  Problems.  We  shall  content  ourselves 
with  pointing  out  the  method  of  solution  of  some  very  simple 
problems  in  determining  the  constants  in  the  equation  of  a 
harmonic  curve.  The  treatment  of  empirical  data  problems 
in  which  the  points  representing  a  given  table  of  values  appear 
to  lie  on  a  harmonic  curve  furnishes  sufficient  material  for  a 
book  involving  considerable  higher  mathematics. 

Example.     Determine  the  constants  a  and  b  if  the  graph  of 

y  =  a  sin  X  •{- b  sin  2x 

is  to  pass  through  the  points  (7r/6,  1  +  a/3/2)  and  (7r/4,  1  +  V2). 

Since  these  points  are  to  lie  on  the  graph  their  coordinates  must  satisfy 


given  equation. 

Hence  1+  -^  =  -a  +  —b, 

l+V2  =  ^a  +  6. 

Solving  these  equations  for  a  and  6,  a  =  2  and  6  =  1.    Hence  the  required 
equation  is  2/  =  2  sin  x  +  sin  2x. 


I                          --^    -          - 

v"-^ 

/       \^2sinx +feir  2j; 

-2           -/■          ^  ^ 

t      ^^V>^ 

i-^^      S  " 

1      //                    \       \ismx 

\/^    "^           X       V        A^^^ 

7^            \  \  -,      X 

jj        \       ^   \  /         sp^-^ 

l             ^        '^^^SS          XX             £ 

0                 i          \     2                sA'-v    '           ^5               e  <2Tr 

V                 '      X      5              \                  1 

^         Z    ^      ^       \        27 

■--1                        ^-.^          ^     _5          '^^y^ 

^   ^           21 

i     5  ^     A^t 

^    ^  ^' 

■:2                                                                                                                  VV^^          7 

V      L 

^    J 

^'^ 

Fig.  208. 

If  several  points  are  given  which  appear  to  lie  on  or  near  the 
graph  of  an  equation  of  the  form  given  in  the  example,  we  may 
determine  a  pair  of  values  of  a  and  b  from  each  pair  of  points.  If 
the  values  of  a  agree  closely,  and  also  those  of  b,  by  using  the 
average  value  of  the  a's  and  that  of  the  6's  we  obtain  a  good  ap- 
proximation of  the  law  connecting  the  coordinates  of  the  points. 


362  ELEMENTARY  FUNCTIONS 

EXERCISES 

1.  Show  that  the  equation  y  =  a  sin  (bx  +  c)  may  be  put  in  the  form 

y  =  A  smbx  +  B  cos  bx, 

where    A  =  a  cos  c        and        B  =  a  sin  c. 

2.  If  2/  =  A  sin  bx  +  B  oos  bx,  find  the  values  of  a  and  c  for  which  the 
given  equation  has  the  form  y  =  a  sin  {bx  +  c). 

3.  Show  that  y  =  a  Bin  {bx  +  c)  may  be  put  in  the  form  y  = 
o  cos  {bx  +  c'),  where  c'  =  c  -  x/2. 

4.  On  the  same  axes  sketch  the  graphs  of 

(a)  sin  X,  sin  ^x,  3  sin  Ix. 

(b)  cos  X,  cos  2x,  I  cos  2x. 

(c)  sin  X,  sin  ttx,  4  sin  irx. 

6.  Determine  the  phase,  period,  and  amplitude  of  each  of  the  following 
functions,  and  sketch  the  graph. 

(a)  3  sin  irx.  (b)  2  sin  irx/2.  (c)  |  sin  ttx/S. 

(d)  2  sin  {ttx  +  tt).  (e)   |  sin  (rc/2  +  x/2).         (f)  2  sin  (x  -  1). 

•    (g)  3  sin  {ttx/S  -  tt).      (h)  I  sin  (7ra:/2  +  x/2).      (i)   4  sin  {irx  -  tt/S). 

6.  Sketch  the  following  compound  harmonic  curves  by  the  addition  of 
ordinates. 

(a)  sin  a;  +  2  sin  2x.  (b)  2  sin  x  +  sin  x/2. 

(c)  3  sin  2x  -  sin  x/2,  (d)  §  sin  x  +  sin  3x. 

(e)  sin  TTX  -  I  sin  7rx/2.  (f)    2  sin  ttx/S  +  |  sin  ttx. 
(g)  2  sin  27ra;/3  +  ^  sin  27rx.  (h)  1.4  sin  .7x  +  .3  sin  1.2x. 

(i)    2  sin  {ttx/ 4:  +  7r/2)  +  |  sin  (27ra;  +  tt). 

7.  Find  the  symmetry,  intercepts,  and  the  points  of  maxima,  minima, 
and  inflection  for  each  of  the  curves  below.  Sketch  the  curve,  and  find 
the  area  of  the  arch  to  the  right  of  the  origin. 

(a)  2  sin  a;  +  sin  2x, 
<b)  sin  X  -  sin  §x. 
(c)  sin  2x  +  cos  2x. 

Note.  Simple  Harmonic  Motion.  Let  a  radius  of  a  circle,  OP,  revolve 
uniformly  at  the  rate  cc,  in  radians  per  second  (see  Example  1,  Section  128), 
and  let  M  be  the  projection  of  P  on  a  fixed  diameter.  The  motion  of  M 
is  called  simple  harmonic  motion.  The  point  M  moves  back  and  forth 
along  the  diameter,  making  one  complete  oscillation  for  every  revolution 
of  OP. 

Let  the  fixed  diameter  be  chosen  for  the  a>axis,  and  suppose  that  P 
starts  on  the  positive  x-axis.  Then  the  abscissa  of  P,  which  gives  the 
position  of  M  at  any  time,  is  s  •»  a  cos  o)t,  where  a  is  the  radius  of  the  circle. 


PROPERTIES  OF  TRIGONOMETRIC  FUNCTIONS      363 

If  P  starts  at  the  point  for  which  OP  makes  an  angle  of  Bq  with  the  x-axis 

then  s  =  a  cos  (co<  +  ^o). 

If  the  diameter  on  which  M  moves  is  taken  for  the  2/-axis,  the  distance 

OM  is  given  by 

5  =  a  sin  (co<  +  do). 

8.  A  point  moves  according  to  one  of  the  laws  below.  Find  the  time 
of  one  oscillation,  and  when  and  where  the  distance,  velocity,  and  accelera- 
tion have  maximum  or  minimum  values.     Describe  the  motion. 

(a)  s  =  8  sin  2t.  (b)  s  =  10  sin  \t.  (c)  s  =  2  sin  irt. 

(d)  s  =  4  cos  xV3.      (e)  s  =  sin  (3i  +  7r/2).      (f )  s  =  2  sin  {irt/Z  +  7r/4) . 

9.  If  s  =  o  sin  {(Jit  +  ^o)  show  that  (a)  the  acceleration  is  proportioned 
to  the  distance,  (b)  that  the  acceleration  and  distance  have  maximum  or 
minimum  values  when  the  body  is  at  rest,  (c)  that  the  body  is  at  the  cen- 
ter of  its  motion  and  has  no  acceleration  when  it  is  moving  fastest. 

10.  The  wind  has  left  long  swells  moving  along  the  surface  of  a  lake 
when  a  slight  breeze  produces  a  ripple.  The  equations  of  cross-sections 
of  the  waves  before  and  after  the  breeze  are  y  =  2  sin  a:/4  and  y  = 
2  sin  x/4  +  \  sin  2x.     Plot  both  curves  on  the  same  axes. 

11.  Determine  the  constants  a  and  h  if  the  graph  of  the  given  function 
is  to  pass  through  the  given  points.     Sketch  the  graph. 

(a)  asmx  +  h  sin  x/2,  (7r/2,  3  -  V2),  (tt,  -  2). 

(b)  a  sin  xJ2  +  h  sin  2x,  (tt/S,  1  -  \/3/4),  (27r/3,  V3  +  V'3/4). 

(c)  a  sm  a;  +  6  sm  2x,  (.157,  .309),  (.332,  .632),  (.611,  1.04). 

(d)  a  sin  X  +  5  cos  x,  (.122,  .570),  (.349,  1.20),  (.646,  1.47). 


MISCELLANEOUS   EXERCISES 

1.  A  and  B  are  two  points  on  the  opposite  sides  of  a  hill  which  are  to 
be  connected  by  a  tunnel.  Both  points  are  visible  from  a  third  point  C 
which  is  1356  feet  from  A  and  2582  feet  from  B.  li  AACB  =  47°.34 
and  if  the  angles  of  elevation  of  A  and  B  &t  C  are  respectively  12°.35  and 
9°. 82,  find  the  length  of  the  tunnel  and  the  inclination  of  the  tunnel. 

2.  Construct  a  table  of  analogous  properties  (see  Section  56)  of  the 
functions  e*,  log  x,  and  sin  x. 

3.  Contrast  the  formulas  for  6"+*  and  tan  (o  +  h)  by  expressing  each 
entirely  in  terms  of  the  notation /(x) . 

4.  Find  approximate  values  of  the  real  roots  of  the  following  equations 
(see  Exercise  12,  page  261). 

(a)  sin  2rc  -  X  =  0.  (b)  cos  ^x  -  x^  =  0. 

(c)  sin  3x  -  e~*  =>  0.  (smallest  positive  root.) 

(d)  cos  2x  -  2x  =  0.  (e)  sin  ^x  -  log  x  =  0. 
(f)   2  cos  2x  -  e*  =  0.                                           (g)  tan  |x  -  x2  =  0. 


364  ELEMENTARY  FUNCTIONS 

6.  To  find  the  distance  between  two  towers  on  opposite  sides  of  a  river 
a  base  line  300  feet  long  was  laid  off.  The  following  angles  were  measured, 
C  and  D  representing  the  towers  and  AB  the  base  line:  AABC=  35°.24, 
ABAD  =  29°.61,  AABD  =  108°.47,  ABAC  =  97°.59.     ComputaCD. 

6.  Find  the  number  of  acres  in  a  field  in  the  form  of  a  quadrilateral 
ABCD  if  AB  =  S15  yards,  ZZ)AC  =  123°.6,  ZCAB  =  54°.68,  ZABD  = 
65°.23,  and    ZABC  =  112° A. 

7.  If  a  projectile  is  fired  at  an  angle  d  with  the  horizontal  with  an  initial 
velocity  of  vo,  the  equation  of  its  path  is 

y  =  X  tan  a  - 


cos^  d 

Find  the  range  as  a  function  of  20,  and  the  value  of  6  which  gives  the 
maximum  range. 

8.  A  corridor  turns  at  right  angles.  It  is  8  feet  wide  on  one  side  of  the 
.turn  and  6  feet  wide  on  the  other.  How  long  a  beam  can  be  carried  hori- 
zontally around  the  corner? 

9.  The  path  of  a  point  on  the  rim  of  a  wheel  rolling  along  a  level  road 
is  given  by  the  equations 

X  =  a(ojt  —  sin  o)t),        y  =  o(l  -  cos  oot), 

where  a  is  the  radius  of  the  wheel,  co  is  the  angular  velocity  of  the  wheel 
about  the  axle,  and  t  is  the  time.  Plot  the  equation  of  the  path  taking 
t  =  7r/6co,  tt/Sco,  7r/2aj,  etc.  Find  the  velocity  of  the  point  at  any  time,  and 
its  maximum  and  minimum  values.  Where  is  the  point  when  it  is  moving 
slowest?  fastest?  A  bit  of  mud  is  thrown  from  the  highest  point  of  a  wheel 
on  an  automobile.  Compare  the  velocity  with  which  the  mud  leaves  the 
wheel  with  that  of  the  automobile. 

10.  If  a  is  the  angle  between  two  lines  whose  slopes  are  mi  and  mz, 

show  that 

mi  —  mi 


tan  a 


1  +  Wi  mj 


CHAPTER  IX 
THEORY   OF   MEASUREMENT 

131.  Statistical  Methods,  The  object  of  an  experiment  is 
to  replace  a  complex  system  in  which  a  nmnber  of  causes  are 
operating  by  a  simpler  system  in  which  causes  are  controlled 
and  allowed  to  vary  only  at  the  will  of  the  investigator.  In 
physics,  for  instance,  after  the  isolation  of  two  related  vari- 
ables, an  endeavor  is  made  to  determine  the  relation  between 
the  variables  and  to  express  this  relation  in  the  form  of  a 
function  y  =/(x). 

In  some  sciences  it  is  difficult  or  impossible  to  isolate  two 
related  variables  from  the  complex  system  in  which  they 
odcur.  In  such  cases  the  variation  of  each  variable  is  observed 
independently  of  the  others. 

Statistics  has  to  deal  with  variables  affected  by  a  number  of 
causes.  Some  of  the  methods  of  statistics  which  have  been 
developed  for  the  analysis  of  such  variations  are  discussed 
briefly  in  this  chapter,  three  of  the  principal  ends  sought 
being  the  determination  of 

(1)  An  average  value  which  will  represent  the  values  of  a 
variable. 

(2)  A  measure  of  the  variability  of  the  items  with  respect  to 
the  average  value  chosen. 

(3)  A  measure  of  the  extent  of  the  relationship  between  two 
variables  which  are  associated. 

The  first  few  sections  are  given  to  some  essential  prerequi- 
sites. To  illustrate  the  fundamental  principle  on  which  the 
following  sections  are  based,  consider  the 

Example.  There  are  five  pitchers  and  three  catchers  on  a  bass  ball 
team.    In  how  many  ways  can  a  battery  be  chosen  for  a  particular  game? 

365 


366  ELEMENTARY  FUNCTIONS 

The  position  of  the  pitcher  can  be  filled  in  5  ways,  and  with  each  of  these 
there  is  a  choice  of  3  catchers.  Hence  the  two  positions  together  can  be 
filled  in  5  X  3  =  15  ways. 

Fundamental  Principle,  If  one  act  can  he  done  in  A  ways 
and  a  second  act  in  B  ways,  the  total  number  of  ways  in  which 
the  two  acts  may  be  performed  in  succession  is  A-B. 

The  following  theorem  is  a  generalization  of  the  fundamental 
principle. 

Theorem.  If  one  act  can  be  done  iri'A  ways,  a  second  in  B 
ways,  a  third  in  C  ways,  and  so  on,  they  can  all  be  done  together 
in  the  order  stated  in  A-B  C-   .  .  .  ways. 

EXERCISES 

1.  If  two  coins  are  tossed,  in  how  many  ways  can  they  fall? 

2.  If  two  dice  are  thrown,  in  how  many  ways  can  they  fall? 

3.  A  committee  of  three  is  chosen  from  seven  physicians,  eight  lawyers, 
and  twelve  business  men,  so  that  each  group  is  represented.  In  how  many 
ways  can  the  committee  be  chosen? 

4.  If  two  dice  and  three  coins  are  tossed,  in  how  many  ways  can  they 
fall? 

132.  Permutations.  Definition.  Each  arrangement  which 
can  be  made  of  all  or  part  of  a  number  of  things  is  called  a 
permutation. 

Thus  the  permutations  of  the  letters  a,  b,  c,  taken  three  at 
a  time  are  abc,  acb,  bac,  bca,  cab,  cba,  and  their  permutations 
taken  two  at  a  time  are  ab,  ba,  ac,  ca,  be,  cb. 

The  number  of  permutations  of  n  different  things  taken  r 
at  a  time  is  denoted  by  nPr- 

Theorem.  The  number  of  permutations  of  n  things  taken  r 
at  a  time  is 

„Pr  =  n(n  -  1)  (n  -  2)  .  .  .  (n  -  r  +  1). 

There  are  r  places  to  fill  and  n  things  from  which  to  choose. 
For  the  first  place  there  is  a  choice  of  n  things.  The  second 
place  can  be  filled  with  any  one  of  the  remaining  n  -  1  things. 
The  third  can  be  filled  in  n  -  2  ways,  and  so  on. 

For  the  rth  place  there  is  a  choice  ofn-  (r-l)orn-r  +  l 
things. 


THEORY  OF  MEASUREMENT  367 

Hence  by  the  theorem  of  Section  131, 

nPr  =  n(n  -  1)  (n  -  2)  .  .  .  (n  -  r  +  1), 

Corollary.  The  number  of  permutations  of  n  things  taken  all 
at  a  time  is 

„P„  =  n{n-l)  (n  -  2)  .  .  .  2  x  1  =  nl 

The  symbol  n !  is  read  factional  n,  and  represents  the  product 
of  all  the  integers  from  1  to  n  inclusive. 

Example,  (a)  In  how  many  ways  can  the  letters  of  the  word  triangle 
be  arranged?  (b)  How  many  of  the  arrangements  will  begin  with  the 
letters  tri  in  this  order?  (c)  How  many  arrangements  will  have  the 
letters  tri  together  in  any  order?  (d)  How  many  arrangements  con- 
sisting of  three  letters  can  be  made  from  the  word  triangle? 

(a)  Here  n  =  8  and  r  =  8,  and  hence  the  number  of  arrangements  is 
gPs  =  8!  =  40,320. 

(b)  The  first  three  places  are  filled  and  there  remain  five  letters  to  be 
permutated  in  five  places,  hence  the  number  of  arrangements  is  sPs  =  120. 

(c)  The  letters  tri  can  be  permuted  as  a  group  with  the  remaining  five 
letters,  and  then  the  three  letters  tri  be  permuted  within  their  own  group. 
Hence  the  number  of  arrangements  is  aPs  X  ePe  =  3!  6!  =  4320. 

(d)  Here  n  =  8  and  r  =  3;  hence  the  number  of  arrangements  is  gPa  = 
8x7x6  =  336. 

EXERCISES 

1.  In  how  many  ways  can  five  flags  of  different  colors  be  arranged 
five  in  a  line?     Three  in  a  line? 

2.  How  many  arrangements  of  all  the  letters  of  the  word  English  will 
t)egin  with  a  vowel,  and  end  with  a  consonant?  How  many  will  have  the 
vowels  together? 

3.  How  many  different  numbers  less  than  1000  can  be  formed  from  the 
digits  1,  2,  3,  4,  5,  6? 

4.  In  how  many  ways  may  first  and  second  prizes  be  awarded,  if  there 
are  12  competitors  in  a  race? 

133.  Combinations.  Definition.  A  group  of  things  which 
is  independent  of  the  order  of  the  elements  is  called  a  com- 
bination. 

The  combinations  of  a,  b,  c,  taken  two  at  a  time  are  ab,  ac, 
be. 

The  selection  6a  is  a  different  permutation  from  ab,  but  the 
same  combination. 


368  ELEMENTARY  FUNCTIONS 

The  six  permutations  of  the  letters  ahc  taken  three  at  a  time, 
namely,  abc,  acb,  bac,  bca,  cab,  cba  are  different  arrangements 
of  one  combination. 

The  number  of  combinations  of  n  things  taken  r  at  a  time  is 
denoted  by  nCr- 

Theorem  1.  The  number  of  combinations  of  n  things  taken 
T  at  a  time  is 

n(n  -  1)  (n  -  r)  .  .  .  (n  -  r  +  1) 

nCr-  ^^— 

Each  combination  of  r  things  chosen  from  the  n  things  can 
be  arranged  in  r!  ways. 

Therefore  all  the  combinations  can  be  arranged  in  r  !„(7r  ways. 

But  this  is  the  number  of  ways  in  which  n  things  taken  r 
at  a  time  can  be  arranged,  so  that 

r\nL>r   =   ni^r 
Pr 


r  —        I 

rl 


n^r 


_  ^(^  -  1)  •  -  ♦  (n  -  r  +  1) 
r\  ' 

Theorem  2.  The  number  of  combinations  of  n  things  taken  r 
at  a  time  is  the  same  as  the  number  of  combinations  of  n  things 
taken  n  —  r  at  a  time. 

Multiplying  numerator  and  denominator  of  the  formula 
for  nCr  by  (n  -  r) !  we  have 

_  n(n  -  1)  .  .  .  {n  -  r  +  1)  (n  -  r)(n  -  r  -  1)  .  .  .  2-1 
"^'■~  r\{n-r)\ 

n\ 
r\{n  —  r)\ 

By  Theorem  1,  we  have 

^       _  n(n  -  1)  .  .  .  (n  -  (n  -  r)  +  1)  _  n(n  -  1)  .  .  .  (r+  1) 
'^^"-'"  "  {n-r)\  ~  {n-r)l 

Multiplying  numerator  and  denominator  by  r! 

n(n  -  I)  .  .  .  (r  +  l)r(r  -  1)  ...  1  ?i! 


»C  n—r    — ' 


{n  —  r)\r\  {n  —  r)\r\ 


THEORY  OF  MEASUREMENT  369 

Example  1.  A  committee  of  5  is  to  be  ciiosen  from  7  lawyers  and  6 
physicians.  How  many  committees  will  contain  (a)  just  3  lawyers, 
(b)  at  least  3  lawyers? 

(a)  The  number  of  ways  o^  selecting  three  lawyers  is 

The  number  of  ways  of  selecting  two  physicians  is 

Hence  the  number  of  committees  which  will  contain  exactly  three 
lawyers  is 

7C3  X  6C2  =  35  X  15  =  525. 

(b)  At  least  three  lawyers  are  present  in  each  committee  of  the  types: 
3  lawyers  and  2  physicians,  4  lawyers  and  1  physcian,  and  5  lawyers. 

Hence  the  number  of  committees  which  contain  at  least  three  lawyers  is 

7C3  X  6C2  +  7C4  X  eCi  +7  Cfi  =  756. 

Example  2.  How  many  arrangements  can  be  made  consisting  of  two 
vowels  and  three  consonants  chosen  from  the  letters  of  the  word  triangle? 

The  vowels  may  be  selected  in  3C2  =  3  ways. 

The  consonants  may  be  selected  in  sCs  =  10  ways. 

Hence  the  total  number  of  selections  consisting  of  two  vowels  and  three 
consonants  is  3C2  X  5C3  =  30. 

Each  of  these  selections  can  be  arranged  in  5!  ways.  Hence  the  required 
number  of  arrangements  is  5!  3C2  X  sCs  =  3600. 


EXERCISES 

1.  How  many  alloys  can  be  made  from  thirty  of  the  known  metals 
chosen  two  at  a  time  counting  one  alloy  only  for  each  pair  of  metals? 
Solve  the  problem  if  there  are  three  metals  in  each  alloy. 

2.  In  how  many  ways  can  a  basket  ball  team  be  selected  from  9  candi- 
dates? If  A  plays  center  in  every  combination,  in  how  many  ways  can 
the  team  be  chosen? 

3.  How  many  arrangements  can  be  made  of  3  vowels  and  4  consonants 
chosen  from  5  vowels  and  8  consonants? 

4.  How  many  straight  lines  are  determined  by  (a)  5  points  no  3  of  which 
are  in  the  same  straight  line?  (b)  n  points,  no  3  of  which  are  in  the  same 
straight  line? 

5.  In  how  many  ways  can  a  baseball  nine  be  chosen  from  13  candidates, 
provided  A,  B,  C,  D  are  the  only  battery  candidates,  and  can  play  in  no 
other  position? 


370  ELEMENTARY  FUNCTIONS 

6.  In  how  many  ways  can  a  committee  of  5  be  chosen  from  7  democrats 
and  7  republicans,  so  that  there  will  be  (a)  three  democrats,  (b)  no  more 
than  three  democrats,  (c)  at  least  three  democrats  on  the  committee? 

134.  The  Binomial  Expansion.  In  finding  the  product  of 
the  binomial  factors  {x  +  ai)(x  +  a2){x  +  as),  each  partial 
product  is  obtained  by  choosing  one  and  only  one  term  from 
each  factor  and  multiplying  the  three  terms  together.  The 
sum  of  the  partial  products  gives  the  desired  product. 

There  is  only  one  term  containing  x^,  since  the  three  x^s  can 
be  chosen  from  the  three  factors  in  but  one  way. 

The  terms  of  the  product  containing  x^  are  obtained  by  choos- 
ing X  from  two  of  the  factors  and  an  a  from  the  third  factor, 
which  gives  the  partial  products  x^ai,  x^a^,  x^as.  The  number 
of  such  terms  will  be  the  number  of  ways  we  can  choose  an  a 
from  the  three  a's,  or  3^1. 

To  obtain  the  term  in  x,  we  choose  an  x  from  one  binomial 
and  two  a's  from  the  two  remaining  in  all  possible  ways,  which 
gives  the  partial  products  xaia2,  xaias,  xa2az.  The  number  of 
such  products  is  therefore  3C2. 

There  is  only  one  way  of  choosing  the  three  a's. 

Hence, 
{x  +  ai){x  +  a2){x  +  az)  =  x^  +  (ai  +  ^2  4-  az)x'^ 

+  (01^2  +  aiaz  +  a2az)x  +  aia^fiz- 

If  we  let  ai  =  a2  =  ^3  =  «,  we  have 

{x  +  ay  =  x»  +  3ax2  +  ZaH  +  a^. 

And  since,  from  the  preceding,  the  coefl&cient  of  x^  is  the  num- 
ber of  ways  we  can  choose  an  a  from  the  three  a's,  the  coeffi- 
cient of  X  is  the  number  of  ways  we  can  choose  two  a's  out  of 
three  a's,  and  so  on,  we  can  write  the  expansion  in  the  form 

(x  +  af  =  x^  +  sCiax^  +  zC^a^x  +  zCza\ 

In  a  similar  manner  it  can  be  shown  that  when  w  is  a  positive 
integer,  we  have 

The  binomial  expansion: 
(x  +  c)"  =  X"  +  wCiCX"-!  +  „  Cio'x"-^  +   .  .  . 


THEORY  OF  MEASUREMENT 
n{n  -  1) 


371 


where  nCi  =  1,  nC2 


21 


n(n  -  1)  .  .  .  (n  -  r  +1) 

7i  ' 


nCr 

The  values  of  the  coefficients  are  given  in  the  following  table  for  several 
values  of  n.    The  table  is  called  Pascal's  triangle. 

n.         123456789    lOlr 


1 

1    1 

2 

1    2 

1 

3 

1    3 

3 

1 

4 

1    4 

6 

4 

1 

5 

1    5 

10 

10 

5 

1 

6 

1    6 

15 

20 

15 

6 

1 

7 

1    7 

21 

35 

35 

21 

7 

1 

8 

1    8 

28 

56 

70 

56 

28 

8 

1 

9 

1    9 

36 

84 

126 

126 

84 

36 

9 

1 

10 

1   10 

45 

120 

210 

252 

210 

120 

45 

10    1 

The  coefficients  in  each  row  in  the  table  may  be  calculated  from  those  in 
the  preceding  row  by  the  following  rule: 

In  any  row  add  to  a  coefficient  the  following  coefficient  and  place  the 
sum  below  the  latter. 

As  an  application,  consider  the 

Theorem.     The  total  number  of  combinations  of  n  things  taken 
one  at  a  time,  two  at  a  time,  and  so  on  up  to  n  at  a  time  is  2"—  1. 
In  the  binomial  expansion  for  {x  +  a^  let  x  =  a  =  1. 

Hence  (1  +  Ij"  =  1  +  nCi  +  nCa  4-  .  .  .  +  nCn. 

Therefore  „Ci  +  nCa  +  .  .  .  +  nCn  =  2"  -  1. 

EXERCISES 

1.  What  is  the  middle  coefficient  in  the  expansion  of  (a  +  6)"?  Using 
Pascal's  triangle  write  the  coefficients  of  the  expansion  (a  +  6)". 

2.  Plot  the  terms  of  the  expansion  (^  4- 1)^^  as  ordinates  at  equal  dis- 
tances along  the  x-axis. 

3.  How  many  compounds  consisting  of  two  elements  could  be  made  from 
eighty-three  chemical  elements?     How  many  consisting  of  three  elements? 

4.  From  eight  men,  in  how  many  ways  can  a  selection  of  four  men  be 
made  (a)  which  includes  two  specified  men?  (b)  which  excludes  two 
specified  men? 


372  ELEMENTARY  FUNCTIONS 

5.  How  many  symbols  would  be  available  for  a  cipher  if  each  symbol 
is  an  arrangement  of  the  letters  a,  b,  in  a  group  of  five.  Thus,  A  =  aaaaa, 
B  =  aaaab,  etc. 

6.  Twelve  competitors  run  a  race  for  three  prizes.  In  how  many  ways 
is  it  possible  that  the  prizes  may  be  given? 

7.  In  how  many  ways  can  a  baseball  nine  be  arranged  if  each  of  the 
nine  players  is  capable  of  playing  any  position?  If  A  must  pitch  and  B, 
C,  D,  play  in  the  outfield?  If  A  or  B  must  pitch,  B  or  C  catch,  and  D, 
E,  F,  play  on  the  bases? 

8.  How  many  dominoes  are  there  in  a  set  from  double  blank  to  double 
six? 

9.  How  many  melodies  consisting  of  four  notes  of  equal  duration  can 
be  formed  from  the  eight  tones  of  the  major  scale?  From  the  thirteen 
tones  of  the  chromatic  scale? 

10.  A  Yale  lock  contains  5  cylinders,  each  capable  of  being  placed  in 
10  distinct  positions,  and  opens  for  a  particular  arrangement  of  the  cylinders. 
How  many  locks  of  this  kind  can  be  made  so  that  no  two  shall  have  the 
same  key? 

11.  The  combination  of  a  safe  consists  of  figures  and  letters  arranged 
on  three  wheels,  one  bearing  the  numbers  0  to  9  inclusive,  another  the 
letters  A  to  M  inclusive,  and  the  third  the  letters  N  to  Z  inclusive.  If  the 
safe  opens  for  but  one  of  these  arrangements,  how  many  different  com- 
binations can  be  used? 

12.  How  many  arrangements  of  all  the  letters  of  the  word  Columbia 
(a)  begin  with  a  vowel?  (b)  begin  with  a  consonant  and  end  with  a  vowel? 
(c)  have  the  vowels  together? 

13.  Show  that  the  number  of  ways  in  which  n  things  can  be  arranged 
in  a  circle  is  (n  -  1)!  In  how  many  ways  can  six  persons  be  arranged  in 
a  line?     In  a  circle? 

14.  Show  that  if  of  n  things  a  are  alike,  h  others  are  alike,  c  others 
are  alike,  etc.,  the  number  of  distinct  permutations,  taken  all  at  a  time,  is 

n! 


alblcl  .. 


16.  How  many  distinct  arrangements  can  be  made  of  all  the  letters  of 
the  word  Mississippi?     International? 

16.  How  many  signals  can  be  made  by  arranging  2  white  flags,  3  red, 
and  1  blue  in  a  row? 

17.  Prove  that  nCr  +  nCr-i  =  n+iCr.     Compare  this  with  the  rule  fo: 
finding  numbers  in  Pascal's  triangle. 

18.  Prove  that  2nCn+r+i  =  2»Cn+r  (     ^~  ^  .),  a  result  which  will  be  used 
later. 


THEORY  OF  MEASUREMENT  373 

19.  How  many  jdiflferent  sums  can  be  formed  with  a  penny,  a  nickle, 
a  dime,  a  quarter,  a  half  dollar,  and  a  dollar? 

20.  A  set  of  weights  consists  of  1,  2,  4,  8,  and  16  ounce  weights.  How 
many  different  amounts  can  be  weighed? 

21.  If  three  coins  are  tossed  in  how  many  ways  can  they  fall?  Solve 
the  problem  for  4  coins. 

22.  If  two  dice  are  thrown  in  how  many  ways  can  they  fall? 

23.  In  how  many  ways  can  the  hands  of  whist  be  dealt? 

24.  How  many  four-figure  numbers  can  be  formed  with  the  digits 
0,  1,  2,  3,  4,  5,  6,  7,  8,  which  are  (a)  divisible  by  two?  (b)  divisible  by 
five? 

135.  Probability.  On  one  of  the  faces  of  a  cube  is  placed 
the  letter  A,  on  two  of  the  faces  the  letter  B,  and  on  the  re- 
maining three  faces  the  letter  C.  If  the  cube  is  thrown  the 
total  number  of  ways  the  cube  can  fall  is  six,  all  of  which  we 
will  assume  are  equally  likely  to  occur.  The  number  of  ways 
that  the  letters,  A,  B,  C,  can  turn  up  are  respectively  one,  two, 
and  three.  In  a  great  number  of  trials  the  letter  A  would 
turn  up  approximately  in  ^th  of  the  total  number  of  trials, 
the  letter  B  in  fths  and  the  letter  C  in  fths.  This  does  not 
mean  that  in  every  set  of  six  trials  A  turns  up  once,  B  twice, 
C  three  times,  but  that  in  the  long  run,  as  the  number  of  trials 
is  increased,  the  frequency  with  which  A,  B,  C  turn  up  ap- 
proximates to  i,  I,  I  of  the  total  number  of  trials. 

Definition.  The  ratio  of  the  number  of  ways  in  which  a 
particular  form  of  an  event  may  occur  to  the  total  number  of 
w^ays  in  which  the  event  can  occur  (all  assimaed  equally  likely) 
is  said  to  be  the  probability  of  the  particular  event. 

Theorem  1.  If  the  probability  that  an  event  will  happen  is  p 
and  the  probability  that  it  will  not  happen  is  q,  then  q  =  1  —  p. 

Let  T  denote  the  number  of  ways  the  event  can  happen, 
F  the  number  of  ways  in  which  the  favorable  form  of  the  event 
can  happen,  and  U  the  number  of  ways  in  which  the  favorable 
form  of  the  event  cannot  happen. 


U 
7 

Since  F  +  t/  =  T, 


Then,  P  =  ^,  q  =  f 


374  ELEMENTARY  FUNCTIONS 

we  have,  dividing  by  T, 

F      U 

or  p  +  q  =  1. 

,\  q  =  1  -p. 

Corollary.  If  the  favorable  form  of  an  event  is  certain  to  happen, 
then  q  =  0  and  p  =  1. 

Example  1.  Find  the  probability  of  not  throwing  a  sum  of  five  with 
two  dice. 

If  the  dice  fall  1,  4,  or  4,  1,  or  2,  3,  or  3,  2,  the  sums  will  be  five.  Hence, 
F  =  4,   T  =  6.6  =  36,  p  =  A  =  i,  9  =  1. 

Hence,  the  probability  of  not  throwing  a  sum  of  five  with  two  dice 
is  |. 

If  a  sum  ot  money  s  is  paid  upon  the  happening  of  an  event 
whose  probability  is  p,  the  product  sp  is  called  the  mathemati- 
cal expectation. 

Example  2.  According  to  the  experience  of  the  American  life  insur- 
ance companies,  of  100,000  children  10  years  of  age,  749  die  within  a  year. 
Neglecting  interest  on  money  and  the  cost  of  administration,  what  would 
be  the  cost  of  insuring  the  life  of  a  10-year  old  child  for  $1000  for  one 
year? 


749 
The  probability  of  a  child  dying  is  p 


100,000 

749 

The  mathematical  expectation  is  •  1000  =  7.49,  and  hence  the  re- 

quired premium  is  $7.49.  That  is,  if  each  of  the  100,000  children  paid 
$7.49,  then  the  sum  $749,000  resulting  would  be  sufficient  to  make  749 
payments  of  $1000  each. 

Example  3.  From  a  bag  containing  8  white  balls  and  6  black  balls, 
5  balls  are  drawn  at  random.  What  is  the  probability  that  3  are  white 
and  2  are  black? 

From  14  balls,  5  can  be  selected  in  uCg  ways. 

From  8  white  balls,  3  can  be  selected  in  gCa  ways,  and  from  6  black  balls, 
2  can  be  selected  in  eCz  ways.  Hence  the  probability  of  drawing  3  white 
and  2  black  balls  is 

8.7.6  6.5 


_      sCs-eCa 

1.2.3  12 

60 

P"  ua  ^ 

14. 13. 12. 11. 10 
1.2.3.4.5 

143 

THEORY  OF  MEASUREMENT  375 

EXERCISES 

1.  From  a  bag  containing  3  white  balls  and  7  black  balls  one  ball  is 
taken  at  random.  What  is  the  probability  that  it  will  be  white?  Black? 
What  is  the  sum  of  these  probabilities? 

2.  From  a  bag  containing  4  white  balls  and  8  black  balls  2  balls  are  drawn 
at  random.  What  is  the  probability  that  they  will  both  be  black?  both 
white?  one  white  and  one  black?     What  is  the  sum  of  these  probabilities? 

3.  If  two  dice  are  thrown  what  is  the  chance  of  throwing  a  sum  of 
seven?  double  sixes? 

4.  If  the  probability  of  an  event  is  ^i,  what  is  the  probability  that  the 
event  will  not  happen? 

5.  Six  persons  are  about  to  seat  themselves  in  a  row.  •  What  is  the  prob- 
ability that  two  specified  persons  will  be  together?     Will  not  be  together? 

6.  If  a  prize  of  $10  is  given  for  drawing  a  red  ball  from  a  bag  containing 
2  red  balls  and  6  white,  what  is  the  value  of  the  expectation? 

7.  If  of  92,637  people  living  at  the  age  of  20  there  are  85,441  living  at 
the  age  of  30,  what  should  be  the  premium  for  insuring  the  life  of  a  person 
of  age  20,  for  $1000  for  10  years,  neglecting  interest  and  administrative 
charges. 

136.  Compound  Events.  Sometimes  it  is  convenient  to 
consider  an  event  as  made  up  of  two  or  more  simpler  events. 
Thus  if  two  balls  are  drawn  from  a  bag  containing  4  white 
balls  and  5  black  balls,  the  double  drawing  may  be  regarded 
as  a  compound  event  made  up  of  two  single  drawings  per- 
formed in  succession. 

The  component  events  into  which  a  compound  event  is 
resolved  are  said  to  be  dependent  or  independent  according  as 
the  occurrence  of  one  does  or  does  not  affect  the  occurrence  of 
the  others. 

If  two  balls  are  drawn  from  the  bag  mentioned  above,  the 
probability  of  drawing  a  white  ball  the  first  time  is  f .  If  a 
white  ball  is  drawn  the  first  time  and  not  replaced,  the  prob- 
ability of  drawing  a  white  ball  the  second  time  is  f .  But  if 
a  black  ball  is  drawn  the  first  time  and  not  replaced,  the  prob- 
ability of  drawing  a  white  ball  the  second  time  is  f .  Hence 
if  the  ball  drawn  the  first  time  is  not  replaced  the  events  are 
dependent,  as  the  form  of  occurrence  of  one  does  affect  the 
occurrence  of  the  other. 

If  the  first  ball  is  drawn  and  then  replaced  the  probability 


376  ELEMENTARY  FUNCTIONS 

of  drawing  a  white  ball  the  second  trial  is  the  same  as  on  the 
first  trial.  The  events  are  independent  as  the  occurrence  of 
one  does  not  depend  on  the  occurrence  of  the  others. 

Several  events  are  said  to  be  mutually  exclusive  if  only  one 
of  the  events  can  happen.  Thus  in  drawing  two  balls  from  the 
bag  mentioned  above  there  are  three  ways  in  which  at  least 
one  white  ball  might  be  drawn.  Both  balls  might  be  white, 
the  first  white  and  the  second  black,  the  first  black  and  the 
second  white.  These  events  are  mutually  exclusive  for  the  oc- 
currence of  one  necessarily  excludes  the  occurrence  of  the  others. 

Theorem  1.  'If  pi,  P2,  -  .  -,  Pn  are  the  probabilities  of  n 
mutvxilly  exclusive  events,  the  probability  that  one  of  these  events 
will  occur  is  equal  to  the  sum  of  their  separate  probabilities. 

Let  T  be  the  total  number  of  ways  in  which  the  event  can 
happen,  Fi  the  number  of  ways  favorable  to  the  first  event, 
F2  to  the  second,  and  so  on.     Then 

Fi  F2  Fn 

Since  each  one  of  the  mutually  exclusive  events  is  favorable, 
the  total  number  of  favorable  ways  is  /^i  +  /^2  +   .  .  .  -f  Fn. 
Therefore  the  probability  that  one  of  them  will  happen  is 

_  Fl  +  i^2  +     .    .    .    i^n   _  ^1     ,    ^     ,  ^. 

P   ~  rp  ~    rn     \     rp     i        '    •    *     rp' 

Hence  p  =  pi  +  P2  -f  .  .  .  +  Pn. 

Example  1.  If  a  cube  has  the  letter  A  on  one  face,  the  letter  B  on  two 
faces,  and  the  numeral  5  on  three  faces,  what  is  the  probability  that  a  letter 
will  turn  up  when  the  cube  is  tossed? 

There  are  two  mutually  exclusive  events  favorable  to  this  contingency. 

(1)  The  turning  up  of  the  letter  A. 

(2)  The  turning  up  of  the  letter  B. 
The  probability  that  A  will  turn  up  is  |. 
The  probability  that  B  will  turn  up  is  |. 

The  probability  that  a  letter  will  turn  up  is  therefore 

Theorem  2.  If  pi  and  p2  are  the  probabilities  of  two  inde- 
pendent events  the  probability  that  both  events  happen  is  the  product 
of  their  separate  probabilities,  pipi. 


THEORY  OF  MEASUREMENT  377 

Let  Ti  and  Fi  be  respectively  the  total  and  favorable  num- 
ber of  ways  the  first  event  can  happen. 

Let  T2  and  F2  be  respectively  the  total  and  favorable  num- 
ber of  ways  the  second  event  can  happen. 

Then  the  total  number  of  ways  the  compound  event  made 
up  of  these  two  events  can  happen  is  T1T2  and  the  number 
of  the  favorable  ways  for  the  compound  event  is  F1F2. 

Hence  the  probability  of  the  compound  event  is 
Fi'  F2       Fi    F2 

In  general  if  pi,  p2,  .  .  .,  Pn  are  the  respective  probabilities 
of  n  independent  events,  the  probability  that  all  the  events 
will  happen  is  the  product  of  all  the  separate  probabilities. 

Example  2.  Find  the  probability  of  throwing  a  six  in  the  first  trial 
only  in  two  throws  of  a  single  die. 

The  probabiUty  of  throwing  a  six  the  first  time  is  |. 

The  probability  of  not  throwing  a  six  the  second  time  is  | . 

These  events  are  independent,  and  the  probability  of  throwing  a  six 
the  first  trial  only  is  if  =  j\. 

Example  3.  If  the  probability  that  A  will  solve  a  problem  is  f  and  the 
probability  that  B  will  solve  it  is  f ,  find  the  probability  that  the  problem 
will  be  solved. 

The  problem  will  be  solved  if  A  and  B  both  succeed,  if  A  succeeds  and 
B  fails,  or  if  A  fails  and  B  succeeds. 

The  probability  that  they  will  both  succeed  is  |4  =  ^|,  since  A's 
success  or  failure  is  independent  of  B's  success  or  failure.  Likewise,  the 
probability  that  A  succeeds  and  B  fails  is  |-f  =  ^3^,  and  the  probability 
that  A  fails  and  B  succeeds  is  |  •  f  =  /g . 

Since  the  three  possible  ways  in  which  the  solution  of  the  problem 
may  occur  are  mutually  exclusive  the  probability  that  the  problem  will 
be  solved  is 

A  second  method  of  solving  the  problem  is  as  follows: 
The  probability  that  the  problem  will  not  be  solved  is  f -f  =  ^^5. 
Hence,  by  Theorem  1,  Section   135,  the  probability  that  the  problem 
will  be  solved  is  1  -  j%  =  |f . 

Theorem  3.  If  there  are  any  number  of  dependent  events  and 
if  pi  is  the  probability  of  the  first,  p2  the  probability  that  when  the 
first  has  happened  the  second  will  follow,  pz  the  probability  that 


378  ELEMENTARY  FUNCTIONS 

when  the  first  and  second  have  happened  the  third  will  follow, 
and  so  on,  then  the  probability  that  the  events  will  occur  in  succes- 
sion in  this  way  is  pi,  P2  .  .  .  Pn- 

The  proof  is  similar  to  that  of  Theorem  2. 

Example  4.  If  a  bag  contains  4  white  balls  and  5  black  balls,  and  if 
2  balls  are  drawn,  what  is  the  probability  of  drawing  a  white  ball  and  a 
black  ball? 

Consider  the  compound  event  as  resolved  into  two  successive  drawings. 

The  probabiUty  of  drawing  a  white  ball  first  is  f. 

The  probabiUty  of  drawing  a  black  ball  second  is  |. 

The  probabiHty  of  drawing  the  balls  in  this  order  is  therefore  |  -f  =  tg. 

But  since  the  balls  may  be  drawn  in  2  ways,  the  probability  of  drawing 
a  white  ball  and  a  black  ball  is  x\  -2  =  f . 

Theorem  4.  If  the  probability  of  the  happening  of  an  event 
in  one  trial  is  p  and  the  probability  of  its  failing  is  q,  the  proba- 
bility of  its  happening  exactly  r  times  in  n  trials  is  nCr  p^q''~^. 

The  probability  that  the  event  will  happen  r  times  and  fail 
n  -  r  times  in  a  given  order  (for  example,  happen  the  first  r 
times  and  fail  the  last  n  -  r  times)  is  p'"g"-''  (Theorem  2). 

The  number  of  ways  this  can  happen  is  the  number  of  ways 
that  the  r  trials  can  be  selected  from  the  n  trials,  which  is 
nCr.  Hence  the  event  can  happen  exactly  r  times  in  nCr 
mutually  exclusive  ways,  and  the  required  probability  is 
therefore  nCrP^q''-^  (Theorem  1). 

Example  5.  What  is  the  probability  of  throwing  six  twice  in  five 
throws  with  a  single  die? 

Here  p  =  i,   g  =  t,  n  =  5,    r  =  2,        and  hence 

P_  r    rM2   m3_  5^4.  1.125  ^625. 
r-^K.2   u;     Kt)   "1.2   36  216      3888 

Theorem  5.  //  the  probability  that  an  event  will  happen  in 
one  trial  is  p,  then  the  probability  that  it  will  happen  at  least  r 
times  in  n  trials  is 

P  =  p"  +  np"-i^i  +  .  .  .  +  nCrP'q''-^. 

An  event  will  happen  at  least  r  times  in  n  trials,  if  it  happens 
n  times,  or  w  -  1  times,  and  so  on  down  to  r  times.  These 
are  mutually  exclusive  events. 


THEORY  OF  MEASUREMENT  379 

HencG;  applying  Theorems  3  and  1,  the  probability  that  an 
event  will  happen  at  least  r  times  in  n  trials  is 

which  reduces  to  what  is  required. 

EXERCISES 

1.  What  is  the  chance  of  making  a  throw  with  two  dice  that  will  be 
greater  than  9? 

2.  A  bag  contains  3  dimes  and  4  quarters.  Three  coins  are  drawn. 
Find  the  value  of  the  expectation.  Suggestion:  Find  the  value  of  the 
expectation  of  drawing  3  quarters,  2  quarters  and  1  dime,  1  quarter  and 
2  dimes,  3  dimes,  and  add. 

3.  Johnny  may  or  may  not  receive  a  birthday  gift  of  $1.00  from  each 
of  five  relatives.     What  is  the  value  of  the  expectation? 

4.  In  a  bag  are  five  white  and  five  black  balls.  If  two  balls  are  drawn 
what  is  the  chance  that  they  will  both  be  white?  Both  black?  one  white 
and  one  black? 

5.  In  a  bag  are  five  white  and  four  black  balls.  If  three  balls  are  drawn 
in  succession  what  is  the  chance  that  they  will  be  white,  black,  white,  in 
this  order,  if  (a)  the  balls  are  not  replaced  each  time,  (b)  the  balls  are 
replaced  each  time. 

6.  A  man  is  sent  8  keys  on  a  ring  for  eight  locks.  What  is  the  proba- 
bility that  he  will  be  able  to  unlock  the  first  lock  with  the  first  key  he 
tries?     With  one  of  the  first  two  keys?     With  one  of  the  first  three? 

7.  Five  cards  are  drawn  from  a  pack  of  52  cards.  Find  the  probability 
that  (a)  there  is  a  pair,  (b)  three  of  a  kind,  (c)  two  pairs,  (d)  three  of  a 
kind  and  a  pair,  (e)  four  of  a  kind,  (f )  a  flush,  (g)  a  straight,  (h)  a  straight 
flush. 

8.  Three  dice  are  thrown.  What  is  the  most  probable  throw?  What 
is  the  probability  of  throwing  exactly  15?     At  least  15? 

9.  If  5  letters  are  chosen  from  a  group  of  4  vowels  and  6  consonants, 
what  is  the  probability  that  a  set  will  begin  with  a  consonant  and  end  with 
a  vowel? 

10.  If  there  were  eight  independent  chances  in  youth  of  growing  one 
inch  above  5  feet,  find  the  probabilities  of  the  statures  from  5  feet  to  5 
feet  8  inches. 

137.  Mortality  Tables.  An  important  application  of  the 
theory  of  probability  is  the  appUcation  to  problems  concerned 
with  the  duration  of  human  life,  such  as  life  insurance,  pen- 
sions, life  annuities,  and  inheritance  tax  laws.  Such  prob- 
lems are  based  on  tables  called  mortality  tables  which  show  the 


380  ELEMENTARY  FUNCTIONS 

number  of  deaths  that  may  be  expected  to  take  place  during 
a  given  period,  among  a  given  number  of  persons  of  a  given 
age.  These  tables  differ  for  different  countries,  different 
races,  in  the  same  country,  different  periods  of  time,  and  for 
the  two  sexes. 

Some  tables  are  constructed  from  the  experience  of  insurance 
companies,  others  from  census  and  vital  statistics  reports. 
The  American  Experience  Table  is  based  on  the  records  of  the 
Mutual  Life  Insurance  Company  of  New  York. 

If  a  mortality  table  is  based  on  a  sufficiently  large  number  of 
observations,  the  difference  between  the  result  furnished  by  the 
tables  and  actual  mortality  is  negligible.  But  this  must  be 
understood  to  apply  to  large  groups  of  people  and  furnishes  no 
surety  to  an  individual. 

The  following  table  is  a  selection  from  the  American  Ex- 
perience Table  of  mortality,  which  gives  the  number  of  people 
Ix  living  at  age  x  out  of  100,000  living  at  the  age  10. 

The  probability  that  a  person  of  age  x 
will  be  alive  at  the  end  of  n  years  is 
denoted  by 

_   Ix+n 

nPx  —      1      ' 

I'X 

The  probability  that  a  person  of  age  x 
will  not  be  alive  at  the  end  of  n  years  is 

'x+n         I'x         i'x+n 


Age 

Number  living 

X 

I. 

10 

100,000 

15 

96,022 

20 

92,637 

25 

89,032 

30 

85,441 

35 

81,822 

40 

78,106 

45 

74,173 

50 

69,804 

55 

64,563 

60 

57,917 

65 

49,341 

70 

38,569 

75 

26,237 

80 

14,474 

85 

5,485 

90 

847 

95 

3 

nQx  =   I   -  nVx  =   I   - 


Ix  h 


Example.  An  inheritance  of  $20,000  is  to 
be  paid  a  boy  10  years  of  age  when  he  becomes 
20.  What  is  the  present  value  of  the  inheri- 
tance, money  being  worth  5%  compounded 
semi-annually? 

If  the  sum  were  certain  to  be  paid  to  the  boy 
in  ten  years  the  present  value  would  be 

P  -  20,000(l  +  '-^y  -  $12,220. 

Since  the  payment  is  contingent  upon  the 
probability  that  the  boy  will  live  to  receive  the 


THEORY  OF  MEASUREMENT  381 

sum,  the  present  value  will  be  the  mathematical  expectation  of  receiving 
$12,220  contingent  upon  the  probability  that  he  will  live  10  years. 

The  probability  that  a  person  10  years  old  will  live  at  least  10  years  is 

^       ^20      92,637 
Hence  the  present  value  of  the  inheritance  is  (12,220)  (.92637)  =  $11,320. 


EXERCISES 

1.  Plot  the  graph  of  the  mortality  table  given  in  Section  137.  By 
means  of  the  graph  estimate  your  own  chance  of  living  to  the  age  of  75. 

2.  A  man  is  45  years  of  age  and  his  son  is  15.  What  is  the  probability 
that  both  will  be  alive  10  years  hence?  What  is  the  probability  that  at 
least  one  will  be  alive? 

3.  A  man  and  his  wife  are  40  and  35  years  old  respectively,  when  their 
child  is  10  years  old.  What  is  the  probability  that  all  will  be  alive  until 
the  20th  anniversary  of  the  child's  birth?  What  is  the  probability  that 
at  least  one  will  survive? 

4.  What  should  be  the  minimum  cost  of  insuring  the  Ufe  of  a  person 
20  years  old  for  $1000  for  five  years?  For  insuring  a,  couple  against  the 
death  of  either  or  both  for  the  same  sum  and  period  if  the  husband  is  30 
and  the  wife  25?    (Neglect  interest,  etc.) 

5.  A  man  makes  a  will  leaving  $40,000  to  his  wife  in  case  she  survives 
him.  A  son  is  to  inherit  the  money  if  he  survives  both  parents.  If  the 
ages  of  husband,  wife,  and  son  are  60,  50,  25  respectively,  and  if  money  is 
worth  5%  compounded  semi-annually,  what  is  the  present  value  of  the 
expectation  (a)  that  the  wife  will  inherit  the  money  in  10  years?  (b)  that 
the  son  will  inherit  the  money  in  15  years? 

6.  In  each  of  the  following  exercises  plot  a  graph  with  the  probabilities 
as  ordinates  at  arbitrary  equal  intervals  along  the  x-axis. 

(a)  A  coin  is  tossed  six  times.  Find  the  probabilities  of  the  various 
ways  in  which  it  can  turn  up  heads. 

(b)  Find  the  probabilities  for  the  various  ways  in  which  two  dice  can 
fall  in  one  throw. 

(c)  In  the  long  run  A  wins  3  games  out  of  4  from  B  at  chess.  Find  the 
probabilities  of  the  various  numbers  of  games  which  A  might  win  in  8 
successive  games. 

(d)  If  a  die  is  tossed  six  times,  find  the  probabilities  of  the  various  ways 
in  which  an  ace  can  turn  up. 

(e)  If  the  quantity  of  a  trait  in  an  individual  of  a  group  is  the  result 
of  a  chance  combination  of  seven  causes,  determine  the  probabilities  of 
the  ways  in  which  the  trait  may  occur. 


382 


ELEMENTARY  FUNCTIONS 


Number  of 

Number  of 

seeds 

apples 

4 

9 

5 

4 

6 

14 

7 

21 

8 

24 

9 

25 

10 

13 

138.  Frequency  Distributions.  At  an  agricultural  experi- 
mental station  110  apples  were  classified  with  respect  to  the 
number  of  seeds  each  contained,  and  the  number  of  apples  in 
each  class  was  determined.    The  results  are  given  in  the  table. 

Thus  9  apples  had  4  seeds  each 
(the  minimum  number  found  in  this 
investigation),  13  apples  had  10  seeds 
each  (the  maximum  number),  while 
25  apples  had  9  seeds  each  (the  most 
frequent  number  occurring). 

Such  an  arrangement  of  the  indi- 
viduals of  a  group,  classified  with  re- 
spect to  some  characteristic  which 
gives  the  number  of  individuals  in 
each  of  the  classes  is  called  a  frequency  distribution  and  the 
table  in  which  the  classes  and  frequencies  are  given  a  frequency 
table. 

A  graphical  representation  of  this  analysis,  called  a  fre- 
quency polygon,  is  obtained  by  plotting  the  magnitudes  of  the 
classes  as  abscissas,  the 
frequencies  as  ordinates, 
and  connecting  the  points 
by  straight  lines  as  in  the 
figure. 

The  magnitude  of  each 
class   in  this  case    is    an 

integer  and  the  class  in-  i     2     s     4     5     e     ?     8     9    lo    n 

tervals  are  said  to  vary 
discretely.  In  case  the 
characteristic  measured  varies  continuously,  as  for  instance  the 
stature  of  a  group  of  men,  the  size  of  the  class  intervals  and 
their  mid-points  are  chosen  arbitrarily. 

The  magnitude  measured  may  vary  discretely  but  by  such 
small  amounts  that  the  number  of  classes  is  so  great  that  the 
variation  of  the  group  with  respect  to  the  characteristic  cannot 
be  easily  determined.  In  such  a  case  the  class  interval  is  en- 
larged by  grouping  the  frequencies  in  two  or  more  adjacent 


,„20 

-s 

|10 

e 
0 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

Number  of  Seeds 

Fig.  209. 


THEORY  OF  MEASUREMENT 


383 


classes  and  associating  the  resulting  frequency  with  the  mid- 
value  of  the  resulting  larger  class. 

Example.  The  grades  in  geometry  of  30  students  were  30,  42,  48,  55, 
60,  64,  68,  71,  72,  74,  75,  76,  77,  77,  78,  78,  78,  79,  80,  82,  82,  83,  84,  85, 
86,  87,  88,  88,  91,  95.  Collect  the  data  in  frequency  tables  with  class 
intervals  of  5%  and  10%. 


Table  1 

Table  2 

Table  3 

Grades 

FreqiLenqj 

Grades 

Frequenqj 

Grades 

Frequency 

28-32 

1 

30-34 

1 

30-39 

1 

33-37 

0 

35-39 

0 

40-49 

2 

38-42 

1 

40-44 

1 

50-59 

1 

43-47 

0 

45-49 

1 

60-69 

3 

48-52 

1 

50-54 

0 

70-79 

11 

53-57 

1 

55-59 

1 

80-89 

10 

58-62 

1 

60-64 

2 

90-100 

2 

63-67 

1 

65-69 

1 

68-72 

3 

70-74 

4 

73-77 

5 

75-79 

7 

78-82 

7 

80-84 

5 

83^87 

5 

85-89 

5 

88-92 

3 

90-94 

1 

93-100 

1 

95-100 

1 

Tables  1  and  2  show  the  data  collected  in  class  intervala^of  5  %  with 
different  mid-points.    In  table  3  the  class  interval  is  10  %. 

The  class  interval  of  1  %  is  too  small  for  an  adequate  presentation  of 
the  data.    Even  class  intervals  of  5  %  leave  some  classes  empty. 

The  class  interval  should  be  chosen  so  as  to  avoid  empty 
classes.  The  smaller  the  number  of  measurements  the  larger 
the  class  interval  should  be,  and  vice  versa.  The  starting  points 
of  the  intervals  are  not  so  material,  but  it  is  convenient  to  take 
them  so  that  the  mid-points  of  the  intervals  are  integers.  In 
age  distributions  the  returns  usually  cluster  about  the  multiples 
of  5,  which  are  taken  as  the  mid-points  of  the  intervals. 

A  second  method  of  representing  frequency  tables  graphi- 
cally is  indicated  in  the  Figs.  210-212  representing  the  tables 
above.  Rectangles  are  constructed  on  the  class  intervals  as 
bases  with  altitudes  equal  to  the  frequencies.  Such  diagrams 
are  called  histograms.  The  area  of  a  histogram  is  the  sum  of 
the  frequencies  times  the  class  interval. 


384 


ELEMENTARY   FUNCTIONS 


If  the  number  of  observations  be  increased  and  the  class 
interval  decreased  the  frequency  polygon  or  histogram  will 


1 — 1 

"^ 

Ll 

1 

1 

1 

1 

1 

1 

z: 

1 1 

30      35      40      45 


55      60      65      70      75 
6  %  Class  Interval 

Fia.  210. 


80   85 


95  100 


30   35   40   45   50   55   60   65   70   75 
6^  Class  Interval 
Fia.  211. 


95     100 


10 

f 

1 

3U  40  50  60  70 

10%  Class  Interval 
Fig.  212. 


80 


90  100 


approach  more  and  more  closely  to  a  smooth  curve.  Such  a 
curve  is  called  a  frequency  curve. 

A  frequency  histogram  is  sometimes  smoothed  by  drawing 
a  curve  through  the  mid-point  of  the  upper  base  of  each  rec- 
tangle in  such  a  way  that  the  area  under  the  curve  is  the  same 
as  the  area  of  the  histogram. 

A  frequency  table  is  smoothed  by  replacing  the  middle 
frequency  of  three  adjacent  frequencies  by  the  average  of  the 
three  frequencies.  The  two  end  values  are  counted  twice 
and  averaged  with  the  adjoining  value. 


THEORY  OF  MEASUREMENT 


Thusii  A,  B,C,D,E,  . 
the  smoothed  values  are 


are  the  values  of  the  frequencies 


2A-\-B    A+B  +  C    B  +  C  +  D 


etc 


Magrnitudes 


Fig.  213. 


If  the  polygons  or  histograms  corresponding  to  successive 
smoothings  of  the  table  are  plotted  we  can  approximate  closely 
to  the  frequency  curve  which  best 
represents  the  data. 

The  types  of  frequency  distribu- 
tions which  are  most  common  are: 

(a)  The  symmetrical  distribution  in 
which  the  frequencies  decrease  to 
zero  symmetrically  on  either  side  of 
a  central  magnitude. 

It  is  found  in  the  distribution  of  errors  in  chemical  and 
physical  measurements  and  in  biological  measurements,  par- 
ticularly the  measurements  of  anthro- 
pology. 

(b)  The     moderately     asymmetrical 

distribution  in  which  the  frequencies 

decrease  more  rapidly  on  one  side  of 

the  value  of  maximum  frequency  than 

on  the  other. 

^'°-2^*-  It  is  the  most    I 

conamon  of  all  the  distributions  occurring     s 

in  all  forms  of  statistics. 

(c)  The  J-shaped  _ 
distribution  in  which 
the  frequency  con- 
stantly increases  or  decreases.  It  is  found 
in  economic  statistics  and  is  characteristic 
of  the  distribution  of  wealth. 

(d)  The  U-shaped  distribution  in  which 
the  frequency  decreases  to  a  minimum  value  and  then  increases. 
It  is  rare.  It  occurs  in  meteorological  statistics  and  in  sta- 
tistics pertaining  to  heredity. 


Magnitudes 


Magnitades 

Fig.  215. 


Magnitudes 
Fig.  216. 


386  ELEMENTARY  FUNCTIONS 

EXERCISES 

1.  The  following  grades  were  obtained  in  a  spelling  test  given  to  third- 
grade  pupils;  65,  85,  55,  60,  100,  23,  92,  74,  73,  75,  76,  94,  82,  17,  10. 
63,  97,  77,  96,  75,  90,  85,  90,  75,  86,  82,  94,  100,  90,  100,  74,  100,  100. 

Construct  the  frequency  polygon,  giving  frequencies  for  class  intervals 
of  5%  arranged  along  the  a;-axis  with  multiples  of  5  (a)  at  the  first  points 
of  the  intervals,  (b)  at  the  middle  points  of  the  intervals.  Construct 
the  smoothed  curves.  Were  the  words  used  a  good  test  of  the  ability 
of  the  pupils  in  spelling? 

2.  Construct  the  histogram  for  the  data  in  Exercise  1,  with  class  in- 
tervals of  10%,  (a)  with  mid-points  at  55,  65,  etc.;  (b)  with  mid-points 
at  60,  70,  etc.  Calculate  the  smoothed  values  in  the  first  table,  and 
draw  a  curve  through  the  ordinates  resulting. 

3.  The  number  of  seeds  per  apple  in  normal  and  aphis-injured  Rome 
apples,  as  determined  by  an  investigation,  are  given  in  the  table. 
Number  of  seeds  per  apple..  0,    1,    2,    3,    4,    5,    6,    7,    8,    9,10,11,12 
Frequencies  of  normal  apples  0,    0,    0,    4,  31,  30,  38,  63,  64,  37,  31,    1,    1 
Frequencies  of  aphis-injured 

apples 5,  14,  20,  29,  44,  37,  28,  44,  38,  25,  13,    3,    0 

Plot  the  histograms  based  on  these  samples.     What  is  the  most  probable 

number  of  seeds  in  the  normal  and  in  the  aphis-injured  apple?     State  two 

dififerences  between  the  two  variations. 

4.  The  following  table  gives  the  degrees  of  cloudiness  of  the  sky  observed 
at  Breslau  during  the  years  1876-1885,  on  a  scale  of  10. 

Degrees  I       0,       1,       2,      3,     4,    5,    6,      7,    8,        9,      10 

Number  of  days,     I     751,    179,    107,  69,  46,  9,  21,  71,  194,   117,  2089 

Plot  the  histogram  and  the  smoothed  curve.  What  is  the  type  of  the 
distribution?  Would  the  arithmetic  average  of  the  measurements  repre- 
sent the  variation  well?  What  is  the  probability  that  a  day  at  Breslau 
will  be  clear?    Very  cloudy? 

5.  Individuals  A  and  B  were  tested  by  hearing  a  series  of  10  letters 
read  at  the  rate  of  1  per  second  and  being  required  to  write  as  many  as 
they  could  remember  in  the  proper  order  of  the  letters,  as  soon  as  the 
reading  was  finished.    Their  scores  were: 


Letters  correct 

4,     5, 

6, 

7, 

8,     9, 

10, 

A  frequencies 

1,     4, 

5, 

8, 

11,     4, 

3, 

B  frequencies 

0,     2, 

4, 

10, 

16,     3, 

1, 

Draw  the  histograms  and  the  smoothed  curves.  Which  is  the  better 
performance?  What  would  be  the  probability  that  A  and  B  could  each 
get  8  correct  in  a  particular  trial? 

6.  The  following  table  gives  the  age  distributions  for  deaths  from 
typhoid,  measles,  scarlet  fever,  influenza,  tuberculosb,  chrculatory  and 
respiratory  diseases  in  New  York  state  in  1916. 


THEORY  OF  MEASUREMENT 


387 


Age 

Typhoid 

Measles 

Scarlet 
fever 

Influenza 

Tubercu- 
losis 

Circula- 
tory 

Respira- 
tory 

diseases 

diseases 

0-1 

2 

293 

9 

143 

371 

165 

4161 

1-2 

3 

404 

26 

58 

301 

37 

1726 

2-3 

3 

136 

31 

39 

194 

35 

638 

3-4 

4 

64 

31 

11 

129 

32 

274 

^5 

5 

25 

30 

9 

78 

39 

149 

5-9 

22 

54 

52 

33 

285 

285 

327 

10-19 

112 

17 

14 

43 

1267 

603 

299 

20-29 

170 

8 

9 

62 

3634 

788 

837 

30-39 

127 

4 

5 

83 

3754 

1252 

1379 

40-49 

76 

12 

123 

2936 

2222 

1761 

50-59 

56 

3 

193 

1748 

3748 

2084 

60-69 

23 

0 

291 

889 

5564 

2257 

70-79 

8 

0 

448 

302 

6464 

2137 

80-89 

0 

0 

365 

45 

3536 

1203 

90-99 

0 

1 

73 

2 

497 

212 

100 

0 

0 

3 

0 

11 

10 

Unknown 

Age 

0 

0 

... 

... 

2 

4 
25,282 

3 

Total 

611 

1,021 

208 

1,977 

15,937 

19,457 

Plot  the  curves  given  by  these  tables,  identify  the  types  of  curves,  and 
state  characteristics  of  the  diseases. 

7.  The  errors  of  observation  in  471  astronomical  measurements  made 
by  Bradley  were  distributed  as  in  the  table,  in  which  the  mid-values  of 
the  magnitudes  of  the  errors  are  given  in  decimals  of  a  second  of  arc. 

Mid-values  of  errors  I  0.5,  1.5,  2.5,  3.5,  4.5,  5.5,  6.5,  7.5,  8.5,  9.5,  10.5 
Frequencies  1 94.     88,    79,    58,    51,    36,    26,    14,    10,     7,   *  8 

Assuming  that  positive  and  negative  errors  occur  with  equal  frequency, 
plot  the  histogram  and  the  frequency  curve.     Discuss  their  forms. 

8.  Discuss  the  form  of  the  frequency  curve  obtained  by  plotting  the 
probabilities  given  by  the  terms  of  the  expansion  (p  +  g)'*,  for  p  =  h 
9  =  l>  ^  =  6,  at  equal  intervals  along  the  x-axis  and  drawing  a  smooth 
curve.  What  is  the  most  probable  number  of  times  the  event  will  occur 
in  6  trials?  If  the  distances  along  the  x-axis  are  multiples  of  unity,  what 
■will  the  area  under  the  curve  from  the  maximum  ordinate  to  the  right 
hand  extent  of  the  curve  represent? 


388  ELEMENTARY  FUNCTIONS 

9.  Plot  the  graph  of  the  frequency  curve  y  =  e~*^.  Determine  approxi- 
mately by  counting  squares  the  abcissa  of  the  point  whose  ordinate  bisects 
the  area  of  the  right  branch  of  the  curve. 

10.  Plot  the  graph  of  the  frequency  curve  2/  =  lo(l+^}    M--j  from 

X  =  -  3  to  X  =  4.     What  is  the  type? 

11.  Plot  the  graph  of  the  frequency  curve  y  =  10x°-^  e'^-'^  from  x  =  0 
to  X  =  10,  and  discuss  the  type. 

139.  Averages.  The  frequency  gives  the  variation  of  the 
magnitude  measured  but  it  is  convenient  in  most  cases  to  have 
a  single  value  to  represent  the  table.  The  representative 
values  or  averages  which  are  commonly  used  for  various  kinds 
of  frequency  distribution  are 

(a)  the  arithmetic  mean, 

(b)  the  median, 

(c)  the  mode, 

(d)  the  geometric  mean, 

(e)  the  harmonic  mean. 

The  arithmetic  mean.  The  word  mean  or  average  alone  is 
generally  understood  to  denote  the  arithmetic  mean.  It  is 
defined  by  the  equation 

.  _  sum  of  the  measurements 
number  of  measurements 

If  there  are  n  measurements  mi,  m^j  .  .  .  m„,  then 

_  mi-\-m2+  .  .  .  +  mn  _  2m 
n  ~    n  * 

where  the  symbol  Sm  is  used  to  denote  the  sum  of  the  m's. 

If  a  measurement  mi  is  obtained  /i  times,  then  the  sum  of 
these  /i  measurements  is  /iWi.  If  the  measurements  Wi, 
m2,  .  ,  .,  mn  occur  with  the  frequencies  /i,  /2,  .  .  .,  /»,  re- 
spectively, then  the  sum  of  all  the  measurements  is 

/iWi  -1-  f^m^  +  .  .  .  +  fnrrin  =  2/m, 

and  the  total  nimiber  of  measurements  is 


THEORY  OF  MEASUREMENT  389 

Hence  the  arithmetic  mean  of  the  measurements  is 

This  equation  defines  the  weighted  arithmetic  mean  of  the 
numbers  mi,  m^,  .  .  .,  mn  with  the  weights  fi,  ji,  .  .  .,  /„. 
It  is  used  whenever  the  numbers  to  be  averaged  are  not  of 
equal  importance. 

Example.  If  the  averages  of  the  weekly  wages  of  three  factories  are 
14,  17,  18  dollars  and  the  number  of  employees  are  450,  360,  670,  re- 
spectively, then  the  average  weekly  wage  for  all  three  factories  is 

^  450  X  14  +  360  X  17  +  670  x  18  ^ 
450  +  360  +  670 

The  calculation  of  the  arithmetic  mean  of  a  frequency  dis- 
tribution is  simplified  by  considering  the  measurements  of  a 
class  as  concentrated  at  the  center  of  the  class  interval,  as- 
suming the  class  interval  as  a  unit  and  applying  an  extension 
of  the  rule  on  page  83,  given  by  the 

Theorem.  If  mi,  m2,  .  .  .,  m„  are  measurements  with  fre- 
quencies fi,  /2,  .  .  .,  fn,  if  A  is  their  true  mean,  E  an  estimated 
value  of  the  mean,  and  di,  d^,  .  .  .,  dn  deviations  of  the  measure- 
ments from  E,  then 

We  have  mi  =  E  +  di,  so  that  firrii  =  fiE  +  fidi, 

and  similarly  /2m2  =  f2E  +  f2d2,  etc. 

Adding 

Ml  +/2W2  +      .    .     .     +fnmn   =   E(fi  +/2  +    .    .    .     +/n) 

H-(M+/2^2+    .    .    .     +fndn). 

Using  the  symbol  2  to  represent  the  summations. 

Xfm  =  E^f+Xfd. 
Dividing  by  2/,        A  =^  =  E +  ^. 


390 


ELEMENTARY  FUNCTIONS 


Corollary.     The   sum   of  the   deviation  from   the   arithmetic 
mean  is  zero. 

^fd  . 


For  if  E  =  A,  then  the  correction  c 


l^d 


is  zero.     Hence 


S/i,  the  sum  of  the  deviation  from  A  =  E,  is  zero. 

The  computation  of  the  correction  c  =^   is  illustrated  in   I 

Example  1.    Find  the  arithmetic  mean  of  the  incomes  of  the  group 
of  famihes  in  the  first  two  columns  of  the  table. 


Number  of 

Deviation  from 

Incomes 

families, 

mean  estimated, 

'd 

(class  intervals) 

/ 

d 

$400-$499 

8 

-4 

-   32 

500-599 

17 

-3 

-   51 

600-699 

72 

-2 

-144 

700-799 

79 

-1 

-    79 

800-899 

73 

0 

0 
-306 

900-999 

63 

+  1 

63 

1000-1099 

31 

+  2 

62 

1100-1199 

18 

+  3 

54 

1200-1299 

8 

+  4 

32 

1300-1399 

8 

+  5 

40 

1400-1499 

1 

+  6 

6 

1500-1599 

6 

+  7 

42 

+  299 

2/ =  384 


2/d  =  -  7 


Estimated  average,  E  =  850. 


=  -  0.0182  class  intervals. 


384 
.-.A  =  850 -^-100 

=  850 -1.8  =  $848.20 

In  the  computation  it  is  assumed  that  the  income  of  the 
families  in  each  class  is  the  income  at  the  middle  of  the  class 


THEORY  OF  MEASUREMENT  391 

interval.  With  this  assumption  an  income  of  $450  is  re- 
ceived by  8  famihes,  of  $550  by  17  famihes,  etc. 

Estimating  the  mean  as  $850,  the  deviations  of  the  incomes 
of  the  successive  classes  from  $850  are  -  400,  -  300,  etc. 
To  simplify  the  computation  we  take  the  class  interval  $100 
as  a  unit,  so  that  the  deviations  d  become   -4,-3,  etc. 

To  find  the  correction,  c,  which  must  be  appUed  to  the 
estimated  mean,  multiply  the  pairs  of  values  of  /  and  d,  ob- 
taining the  column  headed  fd,  find  the  sum  of  the  numbers 
in  the  columns  headed  /  and  fd,  and  divide  the  latter  by  the 
former.  As  $100  was  taken  as  a  unit,  this  quotient  must  be 
multiplied  by  100.  The  correction  to  be  applied  to  the  esti- 
mated mean  of  $850  is  thus  found  to  be  -  $1.8. 

The  median.  If  all  the  measurements  of  a  series  are  ar- 
ranged in  order  of  magnitude,  the  magnitude  of  the  measure- 
ment halfway  up  the  series  is  called  the  median.  The  magni- 
tudes one-quarter  and  three-quarters  up  the  series  are  called 
the  first  and  third  quartiles. 

The  number  of  the  measurement  whose  magnitude  is  the 

71  +  1 

median  is  — —,   where  n  is  the  number  of  measurements. 

The  numbers  of  the  measurements  whose  magnitudes  are  the 
quartiles  are  |(n  +  1)  and  f  (n  +  1). 

Graphically,  the  first  quartile  is  the  abscissa  of  the  point 
whose  ordinate  cuts  off  one-fourth  of  the  area  of  the  frequency 
curve  of  a  distribution  to  the  left  of  it.  The  median  has  one- 
half  the  area  on  either  side,  the  third  quartile  has  one-fourth 
the  area  to  the  right  of  it. 

In  a  frequency  table  where  the  items  are  given  by  classes  it 
is  assumed  for  the  computation  of  the  median  that  the  measure- 
ments are  distributed  uniformly  through  the  class  interval, 
so  that  after  the  class  in  which  the  median  lies  has  been  de- 
termined, its  position  in  that  class  interval  is  found  by  ordinary 
interpolation. 

Example  2.  Find  the  median  and  quartiles  of  the  incomes  of  the 
group  of  families  given  in  Example  1. 


392 


ELEMENTARY  FUNCTIONS 


The  table  is  conveniently  arranged  in  the  form  given. 

Dividing  by  two  the  number  of  terms  plus 


Incomes 
less  than 

Number  of 
families 

/ 

$500 

8 

600 

25 

700 

97 

800 

176 

900 

249 

1000 

312 

1100 

343 

1200 

361 

1300 

369 

1400 

377 

1500 

378 

1600 

384 

one,  we  have 


385 


192.5,  so  that  the  median 


is  the  average  of  the  192nd  and  193rd  terms. 
As  appears  in  the  table,  there  are  176  measure- 
ments in  the  first  four  classes  and  249  in  the 
first  five  classes.  Hence  the  median  lies  in 
the  5th  or  800-899  class,  192.5  -  176  =  16.5 
measurements  from  the  lower  end. 
Hence,  the  median  is 
16.5 


Af.-800  + 


73 


100  =  822.6. 


This   result   means   that   half    of   the   384 
families  have  incomes  less  than  $822.6  and 
half  have  larger  incomes. 
The  quartiles  are  lound  as  follows: 

Since  I  (385)  =  96.25,  the  first  quartile,  Qi,  lies  between  the  96th  and  97th 
terms,  in  the  third  or  600-699  class,  96i  -  25  =  71 J  measurements  from  the 
low^r  end. 


•.  Qi 


600  +  ^.100 


698.9. 


And  since  f  (385)  =  288.75,  Qs  is  in  the  900-999  class,  288f  -  249  -  391 
measurements  from  the  lower  end. 


.-.  C)3  =  900  +  ^-100 


963.1. 


The  mode.  The  mode  is  the  magnitude  of  the  most  fre- 
quent item  of  a  distribution.  It  is  the  abscissa  of  the  maxi- 
mum ordinate  in  a  smoothed  distribution.  The  average  in- 
come, the  average  man  of  the  newspapers  usually  means  the 
modal  income,  the  modal  man. 

In  Example  1,  assuming  that  the  incomes  are  all  at  ihe  mid- 
points of  the  class  intervals,  the  modal  income  would  be  $750. 
Without  this  assumption,  the  modal  income  would  be  some- 
where between  $700  and  $799. 

The  mode  or  modes  of  a  frequency  distribution  may  not  be 
well  defined  for  the  class  intervals  given  in  the  table.  In 
such  cases  the  mode  is  roughly  located  by  grouping  in  larger 
class  intervals  as  shown  in 


THEORY  OF  MEASUREMENT 


393 


Example  3.    The  distribution  of  grades  in  a  class  in  mathematics  are 
given  in  the  following  table.    Locate  the  mode  roughly. 

Frequency 


11 

1 

oL      \ 
or 

1 

oj 

1 

■ 

1 

2 

4 

0 

oJ 

^9 

11 

1 

1 

5 

1 

1 

10 


>10 


The  table  gives  the  frequencies 
for  class  intervals  of  1  %.  There 
is  no  well-defined  maximum  fre- 
quency, and  hence  no  value  of  the 
mode  is  apparent. 

The  numbers  by  the  left-hand 
column  of  brackets  give  the  fre- 
quencies for  the  class  intervals 
95-90,  89-85,  etc.,  but  the  class 
containing  the  mode  is  not  yet 
apparent  so  that  a  class  interval 
larger  than  5%  must  be  chosen. 

The  numbers  by  the  middle 
column  of  brackets  give  the  fre- 
quencies for  class  intervals  from 
99-90,  89-80,  etc.  The  maximum 
frequency,  10,  shows  that  the  mode 
lies  in  the  third  class,  between  79 
and  70. 

The  numbers  by  the  right-hand 
brackets  give  another  grouping  of 
the  frequencies  by  10%  class  in- 
tervals, from  94-85,  84-75,  etc. 
Here  the  mode  appears  in  the 
class  interval  from  84r-75. 

As  the  mode  lies  between  70% 
and  79%  and  also  between  75% 
and  84%,  it  must  therefore  lie  be- 
tween 75%  and  79%. 


In  a  distribution  of  the  symmetrical  type  the  arithmetic 
mean,  the  median  and  the  mode  fall  together.    In  a  moder- 


394 


ELEMENTARY  FUNCTIONS 


Fig.  217. 


ately  asymmetric  group  the  mode,  median  and  mean  lie  in 
the  order  named  with  the  mean  toward  the  longer  branch  of 

the  curve,  as  in  the  figure. 

A  good  approximation  to  the 
value  of  the  mode  is  obtained  by 
the  use  of  the  formula 

Mode  =  Mean 
-  3  (Mean  -  Median). 

This  formula  is  based  on  the 
assumption,  which  has  been  ob- 
served to  hold  approximately  in  a  large  number  of  cases, 
that  the  median  lies  one  third  of  the  distance  from  the  mean 
toward  the  mode.  The  application  of  this  formula  shows 
that  the  modal  income  in  Examples  1  and  2  is 

848.2  -  3(848.2  -  822.6)  =  771.4. 

The  best  method  of  determining  the  mode  is  to  find  the 
equation  of  the  smooth  frequency  curve  that  best  fits  the  data, 

and  then  find  the  ab- 
scissa of  the  maximum 
point  by  equating  the 
derivative  to  zero. 
The  method  is  tedious 
and  difficult. 

A  graphic  method 
of  determining  the 
median  and  the  mode 


Example  4.  Find 
graphically  the  median, 
the  quartiles,  and  the 
mode  of  the  distribution 
in  Example  1. 

Plot  the  data  in  the 
table  in  Example  2,  using 
500,  600,  etc.,  as  abscissas,  the  ordinates  being  respectively  the  number  of 
families  with  incomes  less  than  500,  600,  etc.  For  convenience  in  plotting 
$100  is  taken  as  a  unit  so  that  the  abscissas  are  5,  6,  etc. 


_.jw                                           1 

-S^-                                ±       =«- 

y'" 

^ 

i                        -             '^^ 

A-                           J 

t 

7 

T 

.200 ---^ 

:^:::::::5t:  :::  ::  ::    :: 

zt 

t     - 

7   -     - 

%            U-    -      - 

';) 1 

I     :     .               it 

z 

A        ^ 

M^-t^          ©1     Jlfo    Me                                                                    ^ 

o\       6              7       8    1   9  ,5;  )     u     1:2      :3     11     1&       X 

1     1          L     I  1     1'                    -L 

Fia.  218. 


THEORY  OF  MEASUREMENT  395 

Let  M  and  N  be  the  projections  on  the  y-axis  of  A  and  J5,  the  end  points 
of  the  curve.  Divide  MN  into  four  equal  parts  by  the  points  D', 
E\  F'.  Let  the  perpendiculars  to  the  2/-axis  at  these  points  cut  the 
curve  at  D,  E,  F,  respectively.  Then  the  abscissa  of  D  is  the  median, 
Me  =  820,  approximately,  and  the  abscissas  of  E  and  F  are  the  quartiles, 
Qi  =  700  and  Qa  =  960. 

The  mode  is  represented  graphically  by  the  abscissa  of  the  point  of 
inflection  I.  This  point  may  be  determined  roughly  by  inspection,  or 
by  placing  a  ruler  tangent  to  the  curve  and  rolling  it  along  the  curve  so 
that  it  remains  tangent.  The  direction  of  rotation  of  the  ruler  changes 
when  the  point  of  contact  coincides  with  the  point  of  inflection.  In  this 
figure  the  mode.  Mo,  appears  to  be  about  800. 

The  geometric  mean  is  defined  by  the  equation, 

G  =  Vmim2  .  .  .  Wn. . 

It  is  most  easily  calculated  by  using  logarithms,  since 

J      g  ^  log  mi  +  log  m2  +    .  .  .  +  log  m„ 

The  geometric  mean  is  used  in  averaging  rates  of  increase 
such  as  arise  in  the  study  of  the  growth  of  population,  growth 
of  skill  in  an  individual,  and  relative  changes  in  the  prices  of 
commodities. 

The  trend  of  prices  in  a  series  of  years  is  gauged  by  finding 
the  ratio  of  the  average  price  of  a  commodity  in  any  year  to 
that  in  a  particular  year  which  is  chosen  as  a  base  and  given 
the  arbitrary  value  100.  Numbers  which  are  determined  for 
the  purpose  of  showing  the  trend  in  prices  are  called  index 
numbers. 

If  the  index  numbers  of  three  commodities  for  one  year  are 
a,  h,  c,  and  for  a  second  year  are  ar,  6s,  d,  then  the  ratios  for 
the  three  commodities  are  r,  s,  t,  and  the  geometric  mean  of 
the  ratios  is  \^rst. 

The^eometric  mean  of  the  index  numbers  for  the  first  year 
is  \^ahc  and  for  the  second  is  '\^ahcrst.  The  ratio  of  these 
two  numbers  is  VrsL 

Hence  the  geometric  mean  of  the  ratios  of  the  index  numbers 
of  several  commodities  for  two  years  is  equal  to  the  ratio  of  the 


396  ELEMENTARY  FUNCTIONS 

geometric  mean  of  the  index  numbers  for  the  second  year  to  that 
of  the  first. 

This  property  of  the  geometric  mean  is  the  reason  for  its 
use  in  averaging  index  numbers. 

The  harmonic  mean  is  defined  by  the  equation 

1 


iJ  = 


n\mi      m2        *  *  *       mj 


It  is  used  in  finding  the  average  amount  of  work  performed 
in  a  given  time,  and  the  average  amount  of  a  commodity -pur- 
chased for  a  given  price. 

In  the  equation  v  =  -,  if  s  is  constant  and  t  varies,  then  the 

average  time  for  a  given  distance  would  be  found  by  the  har- 
monic mean. 

For  since  vi='^,V2  =  ^,    .  .  .  Vn  =  r> 

h  I2  In 

the  average  rate  =  ^+^^  +  ---+^--  =  ^(1  +  1  +  .  .  .  +  LV 
^  n  n\ti  ^  ^2  tj 

Hence  the  average  time 


average  rate 


n\ti^t2^      ^tj 

The  mode  determined  as  in  Example  3  can  be  found  roughly 
more  easily  than  the  median  or  mean  can  be  calculated.  The 
calculation  of  the  precise  true  mode  is  more  difficult  than  that 
of  any  of  the  averages.  In  the  case  of  discrete  variation,  as 
for  instance  the  number  of  seeds  in  an  apple,  the  mode  may 
be  the  only  average  that  will  mean  anything,  as  the  values  of 
the  other  averages  are  quite  likely  not  to  occur  in  the  series. 
The  mode  is  the  most  probable  value  of  a  distribution  of  the 
asymmetrical  type,  since  it  is  the  value  that  occurs  most  fre- 
quently in  the  distribution,  and  hence  it  is  the  best  repre- 
sentative value  of  central  tendency.  It  is  the  typical  case. 
It  is  not  useful  if  it  is  desirable  to  give  weight  to  extreme 
variation,  since  all  the  items  of  the  group  do  not  enter  into  its 
determination. 


THEORY  OF  MEASUREMENT  397 

The  median  ranks  after  tho  mode  in  ease  of  determination, 
and  usually  may  be  located  more  precisely. 

If  the  unit  of  measure  is  difficult  or  impossible  to  determine 
and  a  ranking  of  the  items  is  all  that  can  be  attained,  as  in  the 
measurement  of  scholarship,  the  median  is  the  best  repre- 
sentative value.  If  the  values  of  some  of  the  items  are  given 
vaguely,  as  for  instance  if  items  of  an  upper  class  are  given  as 
greater  than  some  value,  the  median  can  be  determined  more 
precisely  than  the  mean. 

All  the  items  enter  into  the  determination  of  the  median 
but  extreme  cases  affect  its  value  but  slightly.  Changes  in 
the  values  of  extreme  cases  would  affect  the  value  of  the  mean 
without  disturbing  the  value  of  the  median. 

The  arithmetic  mean  is  the  average  most  generally  employed. 
It  is  the  most  familiar  of  the  averages,  is  the  most  precise  when 
all  the  items  are  given,  gives  weight  to  extreme  deviations, 
which  is  desirable  in  certain  cases,  and  is  affected  by  every 
item  in  the  distribution.  It  may  be  determined  if  the  ag- 
gregate and  the  number  of  items  are  known  though  knowledge 
of  the  values  of  the  items  is  lacking,  and  conversely  the  ag- 
gregate may  be  determined  if  the  mean  and  the  number  of 
items  is  known.  This  property  is  not  possessed  by  the  other 
averages. 

The  harmonic  mean  and  geometric  mean  are  used  less  fre- 
quently than  the  other  averages  as  they  are  unfamiliar  and 
more  difficult  to  calculate.  The  arithmetic  mean  is  sometimes 
incorrectly  employed  in  place  of  the  harmonic  mean  in  averag- 
ing time  rates,  and  sometimes  incorrectly  in  place  of  the  geo- 
metric mean  in  averaging  rates  of  increase.  If  variations  are 
measured  by  their  ratio  to,  rather  than  by  their  difference 
from,  the  average,  then  the  geometric  mean  is  the  best  average 
to  employ. 

EXERCISES 

1.  Which  average  is  meant  in  the  following:  average  student,  average 
wage  in  an  industry,  average  daily  temperature,  average  stature,  average 
number  of  potatoes  of  a  given  species  in  a  hill,  average  annual  rainfall, 
average  gain  per  year  in  height  of  a  child,  average  ability  in  arithmetic. 


396  ELEMENTARY  FUNCTIONS 

average  rate  of  increase  of  population^  average  time  in  doing  a  piece  of 
work? 

2.  Find  the  mean,  median,  and  mode  of  the  distribution  in  the  example 
in  Section  138. 

3.  Ten  men  in  a  department  can  complete  a  piece  of  work  in  the  fol- 
lowing number  of  minutes  respectively,  45,  50,  60,  60,  60,  65,  65,  70,  75, 
and  85.     What  is  the  average  time  for  the  work? 

4.  The  population  of  a  city  increased  in  a  decade  from  185,000  to 
260,000.     What  was  the  average  annual  rate  of  increase? 

5.  By  the  probable  duration  of  life  of  a  man  m  years  of  age  is  meant 
the  number  of  years  which  he  has  an  even  chance  of  adding  to  his  life. 
By  the  expectancy  of  life  for  a  man  of  m  years  is  meant  the  arithmetic 
mean  of  the  number  of  additional  years  of  life  enjoyed  by  all  men  m  years 
of  age.  Find  the  probable  duration  and  the  expectancy  of  life  of  a  man 
20  years  old  (use  the  table  in  Section  137). 

6.  The  following  table  gives  the  distribution  of  wages  per  week  of  a 
group  of  laborers. 

Wages  (mid-values)     I     1,  3,    5,    7,     9,     11,    13,   15,    17,    19,    21,  23 
Frequency  I     5,  23,  50,  80,  105,  130,  160,  165,  148,  120,  36,     3 

Find  the  mean,  median  and  mode  of  the  distribution.  Which  average 
represents  the  table  best? 

7.  What  is  the  average  age  at  death  for  each  disease  in  the  tables  in 
Exercise  6,  page  387.  What  average  should  be  used?  Determine  by  in- 
spection of  the  tables  which  are  diseases  of  children  and  which  of  adults. 
Which  warrants  being  called  a  vacation  disease?     Why? 

8.  The  frequencies  of  distribution  of  budgets  of  a  group  of  college 
students  with  a  class  interval  of  $50  starting  at  $350  and  running  to  $1800 
inclusive  were  4,  15,  21,  26,  39,  46,  52,  32,  34,  24,  17,  17,  14,  11,  8,  10,  6, 
6,  4,  4,  3,  3,  1,  1,  0,  2,  1,  0,  0,  3.  Calculate  the  median  and  quartiles  al- 
gebraically and  graphically.  Which  average  would  best  represent  the 
distribution? 

9.  In  1913  the  index  numbers  for  steak,  bacon,  chickens,  eggs,  butter, 
milk,  flour,  potatoes,  sugar  were  94,  95,  97,  91,  108,  100,  100,  90,  100.  In 
1918  the  numbers  in  the  same  commodities  were  respectively  131,  179, 
170,  177,  151,  151,  200,  188,  193.  Calculate  the  average  index  numbers 
in  each  year  and  the  relative  increase  in  prices. 

140.  Measures  of  Variability.  Next  in  importance  to 
selecting  an  average  to  represent  a  group  of  measurements  is 
the  determination  of  a  measure  of  the  extent  to  w^hich  the 
other  values  cluster  about  or  are  dispersed  from  the  average 
chosen,  that  is,  a  measure  of  the  variability  of  the  group  with 
respect  to  the  average. 


THEORY  OF  MEASUREMENT  399 

The  average  obviates  the  necessity  of  stating  all  the  measure- 
ments from  which  it  is  derived,  and  a  measure  of  variability- 
is  a  single  number  characterizing  the  deviation  from  the  aver- 
age. A  series  of  measurements  can  thus  be  summarized  by 
two  numbers  which  are  usually  written  in  the  form  a  =*=  rf, 
where  the  first  number  gives  the  average  value  chosen  and  the 
second  gives  a  measure  of  the  deviations  of  the  items  of  the 
series  from  the  average. 

The  principal  measures  of  dispersion  or  variability  of  a 
distribution  from  an  average  are  (a)  the  quartile  deviation, 
(b)  the  mean  deviation,  (c)  the  median  deviation  or  probable 
error,  (d)  the  standard  deviation. 

The  quartile  deviation  is  defined  by  the  equation  Q  =  --^-^ — -* 

where  Qi  and  Q3  are  the  first  and  third  quartiles  respectively. 
It  is  the  simplest  measure  of  deviation  to  calculate. 

The  quartile  deviation  for  the  distribution  of  incomes  in 
Example  2,  Section  139,  is 

^^963.1-698.9^^3^^ 

Hence  the  group  is  summarized  by  the  median  value  of  the 
group  and  the  quartile  deviation  in  the  form  822.6  =*=  132.1. 
This  means  that  approximately  50  %  of  the  incomes  lie  between 
$822.6  -  $132.1  =  $690.5  and  $822.6  +  $132.1  =  $954.7. 

The  mean  deviation  is  the  arithmetic  mean  of  the  numerical 
values  of  the  deviations  from  an  average.  The  method  of 
calculation  for  a  frequency  distribution  is  shown  in 

Example  1.  Find  the  mean  deviation  from  the  arithmetic  mean  of  the 
distribution  of  incomes  in  Example  1,  Section  139. 

In  this  example  the  estimated  mean  is  E  =  850,  the  numerical  value  of 
the  correction  is  c  =  1.82  units  =  0.0182  class  intervals  and  the  true  mean  is 
A  =  848.2. 

The  sum  of  the  numerical  values  of  the  deviations  from  E  is  306  +  299 
«=  605  class  intervals. 

The  deviation  from  A  of  each  of  the  176  items  less  than  A  is  0.0182  class 
intervals  less  than  the  deviation  from  E. 

The  deviation  from  A  of  each  of  the  73  +  135  =  208  items  greater  than 


400  ELEMENTARY  FUNCTIONS 

A  is  0.0182  class  intervals  greater  than  the  deviation  from  E.    Hence  the 
sum  of  the  numerical  values  of  the  deviations  from  A  is 

605  +  208  X  .0182  -  176  x  .0182  =  606. 

Dividing  by  n  =  384,  we  have 

The  mean  deviation  from  A  =  ^^  =  1.6  class  intervals  =160  units. 

The  median  deviation  or  probable  error.  If  all  the  devia- 
tions from  some  one  of  the  averages  are  arranged  in  order  of 
magnitude  without  regard  to  sign,  the  median  deviation  is 
calculated  in  the  same  way  as  the  median  of  the  distribution. 

The  middle  50  %  of  the  items  come  within  the  range  of  the 
median  deviation  if  the  median  is  the  average  used  to  repre- 
sent central  tendency. 

Approximately  50  %  of  the  items  come  within  the  range  of 
the  quartile  deviation,  but  these  cases  are  not  necessarily  the 
middle  50  %  since  the  median  does  not  lie  exactly  halfway 
between  the  quartiles  except  in  a  symmetrical  distribution. 
The  median  deviation  is  usually  called  the  probable  error  and 
will  be  discussed  further  in  Section  142. 

The  standard  deviation  is  defined  by  the  equation 

di'  +  ^2^  +  .  .  .  dr?      2(i2 


or 


2   _ 


n 


where  di,  d2,  .  .  .,  dn  are  the  deviations  of  the  measures 
from  the  average  chosen.  If  the  deviations  occur  with  the 
frequencies  /i,  /2,  .  .  .,  fn,  then  the  standard  deviation  is 


This  measure  of  dispersion  gives  more  weight  to  extreme 
cases  than  the  other  measures  of  variability,  and  while  more 
tedious  to  calculate  is  more  generally  used. 

In  all  deviation  measures  retain  at  most  two  significant- 
figures.     The  reason  for  this  rule  is  as  follows.     Consider  a 
length  I  =  324.57  cm.  with  a  mean  deviation  of  .14  cm.     The 
mean  deviation  indicates  that  the  figure  5  in  Z  in  the  first 
position  after  the  decimal  point  is  uncertain  by  1  unit  and  that 


THEORY  OF  MEASUREMENT  401 

the  next  figure  7  is  uncertain  by  14  units.  The  next  figure 
would  be  uncertain  by  at  least  140  units  and  is  discarded. 

If  the  first  significant  figure  of  the  deviation  is  8  or  9,  the 
figure  in  the  corresponding  position  of  the  average  and  the 
figure  following  are  retained  but  only  one  significant  figure  of 
the  measure  of  deviation  is  retained. 

The  calculation  of  the  standard  deviation  is  simplified  by 
means  of  the 

Theorem.  If  xi,  X2,  .  .  .,  Xn  are  deviations  from  the  arith- 
metic mean  A  of  a  series  of  measurements  with 

frequencies  /i,  /2,  .  .  .,  /«,  if  di,  c?2,  .  .  -,  dn   — I 1 ^-* 

are  the  deviations  from  an  estimated  mean  E,      c *-      ^ 

and  if  A  =  E  +  c,  then  *^         ^ 

Fig.  219.        = 
o"!  =  (T<t  -  cr, 

where  dx  and  Cd  are  the  standard  deviations  of  the  deviations 
from  A  and  E  respectively. 

Since  A  =  E  +  c,  we  have,  by  the  directed  lines  in  the 
figure,  for  the  first  measurement,  di  =  Xi -{■  c. 

Similarly,   d2  =  X2  +  c,  .  .  .,   dn  =  Xn  -\-  c. 

Hence  the  sum  of  the  squares  of  the  deviations  from  E  is 

S/^^  =  S/(a:  +  c)2  =  S/a;2  +  2c'Zfx  +  c^S/. 

But  llfx  =  0,  since  the  sum  of  the  deviations  from  the  mean 
is  zero,  and  hence 

i:fd' =  Xfx'' +  c'l^f.  (1) 

Transposing  and  dividing  by  S/, 

Xf  -   Xf      "" 
or  a-/  =  (7/  _  c2.  (2) 

The  value  of  the  correction  c  is  (Theorem,  Section  139) 

2/ 

Corollary  1.  The  sum  of  the  squares  of  the  deviations  from  the 
arithmetic  mean  A  is  less  than  the  sum  of  the  squares  of  the  devior- 
tions  from  any  other  number  E. 


402 


ELEMENTARY  FUNCTIONS 


For,  from  (1),  S/d^  is  greater  than  S/x^  by  the  positive  num- 
ber c^S/. 

Corollary  2.  The  standard  deviation  of  a  series  of  measure- 
ments is  a  minimum  when  the  deviations  are  measured  from  the 
arithmetic  mean. 

This  follows  from  equation   (2). 

An  application  of  this  theorem  is  made  in 

Example  2.  Find  the  arithmetic  mean  and  the  standard  deviation 
for  the  following  distribution  of  grades  in  geometry. 


Deviation 

Grade 

Frequency 

/ 

from  E 
d 

Id 

fd' 

95^100 

3 

+  5 

15 

75 

90-94 

9 

+  4 

36 

144 

85-89 

12 

+  3 

36 

108 

80-84 

8 

+  2 

16 

32 

75-79 

14 

+  1 

14 
+  117 

14 

70-74 

9 

0 

0 
-    68 

0 

65-69 

17 

-1 

-17 

17 

60-^ 

11 

-2 

-22 

44 

55-59 

2 

-3 

-    6 

18 

60-54 

2 

-4 

-    8 

32 

Below  50 

3 

-5 

-  15 

75 

2/  =  90  2/(i  =  49     2/d'  =  559 

Let  the  estimated  mean  be  -E  =  72.5  and  let  the  class  interval  be  taken 

as  a  unit. 

Sfd         49  49 

The  correction  is  c  =-^  ""  "^  on  ^^^^  intervals  =  ^^  X  5  %  =  2.7  %. 

Hence  the  mean  is  A  =  E  +  c  =  72.5  +  2.7  =  75.2. 
The  standard  deviation  of  the  deviations  from  E  is 


,     2/d2     559 
""^^T  ="90 


Hence 


(r,2  =  (^^  _  c2  =  6.2 


6.2  class  intervals. 

2 


-Ci) 


6.2  -  .29  =  5.91 


.*.  ffx  =  2.4  class  intervals  =  12  %. 
Hence  the  average  grade  in  the  above  distribution  is  75.2  %  and  the  stand- 
ard deviation  is  12  %. 


THEORY  OF  MEASUREMENT  403 

The  absolute  value  of  two  measures  of  variability  may  be 
the  same  and  yet  their  significance  be  quite  different,  for  in- 
stance, if  two  averages  and  their  measures  of  variability  are 
25  ±  5  and  250  =*=  5,  the  first  indicates  greater  relative  vari- 
ability than  the  second. 

A  measure  of  the  relative  variability  of  a  set  of  measure- 
ments is  obtained  by  dividing  the  measure  of  deviation  d  by 

the  average  a.    The  quotient  -  is  called  a  coefficient  of  relative 

variability.     The  quantity  v  =  100  -j,  which  gives  the  ratio  of  the 

standard  deviation  to  the  arithmetic  mean  expressed  as  a  per- 
centage, is  called  the  coeffix^ient  of  variation. 

EXERCISES 

1.  (a)  Find  the  median  grade  and  the  quartile  deviation  of  the  follow- 
ing distribution  of  grades  in  algebra,  the  given  grades  being  at  the  left- 
hand  ends  of  the  intervals. 

Grade  I  55,         60,      65,      70,      75,      80,      85,      90,      95, 

Frequency       I  3,  7,       10,       14,      22,      26,      20,      16,        5, 

(b)  Find  the  arithmetic  mean  and  the  standard  deviation  of  the  dis- 
tribution. 

(c)  Find  the  mean  deviation  from  the  median. 

(d)  Find  the  mode  and  the  median  deviation  from  the  mode. 

(e)  Draw  the  frequency  curve  of  the  distribution  and  show  the  graphi- 
cal significance  of  each  of  the  averages  and  the  corresponding  dispersion. 
Which  pair  of  numbers  best  represents  the  distribution  and  variability? 

2.  By  the  use  of  directed  lines  show  that  the  mean  deviation  from  a 
point  of  reference  of  a  set  of  seven  points  placed  arbitrarily  on  the  x-axis 
is  least  when  the  point  of  reference  is  the  median  point  of  the  set. 

3.  Compare  the  coefficients  of  variability  from  the  arithmetic  mean  of 
a  group  of  men  and  a  group  of  women  measured  with  respect  to  the  num- 
ber of  associations  set  up  by  a  series  of  words. 


Number  of 
associations 

2, 

4, 

6, 

8, 

10, 

12, 

14, 

16, 

18, 

20, 

Frequencies, 
men 

3, 

7, 

25, 

51, 

55, 

38, 

42, 

20, 

8, 

2, 

Frequencies, 
women 

0, 

8, 

9, 

35, 

57, 

66, 

30, 

18, 

10, 

6, 

404  ELEMENTARY  FUNCTIONS 

4.  The  following  table  gives  the  errors  in  minutes  in  the  predictions  ot 
high  water  at  Portsmouth  during  three  months  in  1897.  Find  the  median 
deviation,  the  mean  deviation,  and  the  standard  deviation.  Approxi- 
mately what  fractional  part  of  the  standard  deviation  is  the  median  devia- 
tion? the  mean  deviation? 


Errc 

)rs 

0-5, 

6-10, 

11-15, 

16-20, 

21-25, 

26-30, 

31-35, 

52 

Free 

juencies 

69, 

50, 

25, 

10, 

11, 

7, 

4, 

1 

141.  Equation  of  the  Frequency  Curve  Representing  a 
Sjnnmetrical  Distribution.  The  derivation  of  the  equation  is 
based  on  the  theory  of  probability. 

If  a  coin  is  tossed  four  times,  the  different  ways  in  which  it 
may  fall  are  heads  every  time,  heads  three  times  and  tails  once, 
heads  twice  and  tails  twice,  heads  once  and  tails  three  times, 
and  tails  every  time.  The  probabilities  of  the  different  ways 
in  which  it  can  fall  are  the  terms  of  the  expansion 

1  =  (1  + 1)4  =  (1)4  +  4(1)3(1)1  +  6(1)2(1)2  +  myay  +  (1)4 

=  I-,  (1  +  4  +  6  +  4  +  1) 
=  tV  +  i  +  I  +  i  +  tV- 
Now   consider  the  adjoined  frequency  table  in  which  an 
arbitrary  interval  Ax  is  chosen,  and  in  which  the  frequencies 

-  2Ax,  -  Ax,  0,  Ax,  2Ax     ^^^  ^^^  *^™^  ^^  *^^  expansion 
-^ — - — —     above.     The  frequency  polygon 

^''        ^'    ''    ^'     ^^      (page    405)    plotted    from    this 

table  is  symmetrical  with  respect  to  the  2/-axis. 

The  standard  deviation  for  this  table  is 

2  ^  5/m2  ^  A(-  2Axy  +  K-  ^xY  +  f -0^  +  i(Ax)2  +  A(2Ax)^ 
2/  -^V  + 1  +  1  +  1  +  ^ 

Therefore  a  =  Ax. 

Plotting  the  table,  using  Aa:  =  c  as  the  unit  on  the  a:-axis 
we  obtain  the  upper  frequency  polygon  in  the  figure. 

If  the  coin  is  tossed  six  times,  the  probabilities  of  the  ways 
in  which  it  can  fall  are  given  by  the  terms  of  the  expansion 

(i  +  §)'  =  4-6  a  +  6  +  15  +  20  +  15  +  6  +  1). 


THEORY  OF  MEASUREMENT 


405 


The  standard  deviation  of  the  frequency  table  analogous 
to  the  above  is  found  to  be  cr  =  Vf  Aa;,  so  that  Ax  =  Vf  o-  =  .8(7. 
The  frequency  polygon  for  this  case  is  the  middle  one  in  the 
figure.  It  is  assumed  that  cr  has  the  same  value  in  both  cases, 
so  that  the  value  of  Ax  is  smaller  than  in  the  preceding  case. 


"^ 

^ 

"" 

~" 

~ 

-1 

"" 

~" 

~" 

"" 

, 

y 

- 

~ 

- 

" 

^ 

^ 

^ 

H 

-~ 

^ 

f=2 

^ 

■^ 

^ 

-^ 

^- 

^ 

•Ji. 

~^ 

r 

■^ 

n- 

rl 

-- 

>> 

^vj 

^ 

^ 

' 

:^ 

^ 

rJ 

> 

1 

1 

" 

vj 

^ 

^ 

1 

1 

=5i 

, 

b. 

^ 

" 

1 

1 



_ 

-20- 

-cr 

0 

.l<T',sq  a- 

2<r 

X 

J 

u 

n 

J 

1 

-L 

1 

1 

1 

_ 

Fig.  220. 

If  the  coin  is  tossed  eight  times,  a  similar  procedure  gives 
the  lowest  polygon  in  the  figure.  In  this  case  a  =  v^2Aa;,  so 
that  Ax  =  (T/V2    =  .7(7. 

These  three  cases  may  be  regarded  as  the  particular  cases 
obtained  by  tossing  a  coin  an  even  number  of  times,  2n,  for 
n  =  2,  3,  4.  In  each  case  the  frequency  polygon  is  symmetrical 
with  respect  to  the  y-axis.  As  n  increases  and  a  keeps  the 
same  value,  the  vertices  of  the  polygon  get  closer  together. 

A  comparison  of  the  values  of  a  forn  =  2,  3,  4  shows  that  in 
each  case  2a^  =  n{Axy. 

With  these  preliminaries,  consider  the  expansion 
/I      1\2"       1  / 
'      \2  ^  2/     ^  2^\    "^  ^"^^  "^  ^"^^  "^  *  * 


The  symmetrical  fre- 
quency polygon  in  the 
figure  is  obtained  by  plot- 
ting the  terms  of  this  ex- 
pansion as  ordinates  at  in- 
tervals Ax  along  the  x-axis, 
with  the  maximum  term, 
2nCn  /22",  ou  the  2/-axis. 


+  2nCn  + 
+  2nCn+r+l  +    . 


SnCn 


H~  2nCn4-r 
+  2nC2nj. 


■* r6.x «j 


FiQ.  221. 


406  ELEMENTARY  FUNCTIONS 

Let  n  increase  indefinitely.  We  assume  that  n  increases 
and  Ax  decreases  in  such  a  way  that  n(Axy  always  remains 
equal  to  a  constant  2(t^,  and  that  the  frequency  polygon  ap- 
proaches an  ideal  symmetrical  frequency  curve. 

To  derive  the  equation  of  this  curve,  let  P  and  Q  be  the 
successive  vertices  of  the  polygon  determined  by  the  rth  and 
(r  +  l)st  terms  from  the  middle  term  of  the  expansion  above. 
Then  the  ordinates  of  these  points  are 

2nLn+r  j  ,      a  2n^  n+r+1 

y  =  "2^        and        y  +  Ay  =  -^i;^— 

Since  2nCn+r+l  =  2nCn+r  (    — — T~i  j? 

we  have     y  +  Ay  = 


tnCn+r  U  -  T  71  -  r 


22n      ^_|_y._|_i      ^  n  +  r  +  1 
Hence  ^  J  ^^TTTl  '  ^       y-2r-l 


Ax  Ax  Ax  n  -\-r  +  \ 

The  abscissa  of  P  is  x  =  r  Ax,  and  hence  ?*  =  x--    Substituting 

this  value  of  r,  and  simplifying, 

Ay  _  -y_       2x  4- Ax       _  _  2x  +  Ax 

Ax  ~  Ax  ?iAx  +  x  +  Ax  ~      ^n(Ax)2  4-xAx+ (Ax)2' 

Replacing  n{AxY  by  2o'2,  in  accordance  with  the  assumption 
made  above, 

Ay  2x  +  Ax 


=  -2/, 


Ax  ''2or2  +  xAx+ (Ax)2 

Passing  to  the  limit  as  Ax  approaches  zero,  the  rate  of  change 
of  y  with  respect  to  x  is  seen  to  be 

2x 
D.y^-y^,, 

whence  — --  = ^• 

y  0-2 

D  y 
In  the  integral  calculus  it  is  shown  that  the  integral  of  — ^ 


THEORY  OF  MEASUREMENT  407 

is  loge  y.     Integrating  by  means  of  this  result  we  have 

loge  2/  =  -  ^2  +  loge  C*       or        log.  ^  =  -  ^• 

Hence  ^  --^.  /i\ 

y  ^  Ce  20-2.  U; 

This  is  the  equation  of  the  curve  with  the  constant  of  inte- 
gration still  undetermined.  The  curve  is  called  the  probability 
curve  (see  Fig.  222,  page  409). 

Discussion  of  the  Curve.  Symmetry.  The  curve  is  symmetri- 
cal with  respect  to  the  y-SLxis,  since  the  equation  is  unchanged 
when  X  is  replaced  by  —  a;. 

Intercepts.  The  intercept  on  the  2/-axis  is  (7,  for  if  x  =  0, 
then  y  =  C. 

Asymptotes.  The  a:-axis  is  an  asymptote.  For  y  approaches 
zero  as  x  becomes  infinite,  since  the  exponent  of  e  is  always 
negative  and  increases  numerically  when  x  increases. 

Points  of  Inflection.  To  find  the  points  of  inflection,  we 
assume  without  proof  the  formula  D^e"  =  e"  D^u. 

_£i 
Differentiating  y  =  Ce  2<r* 

we  obtain  the  slope  of  the  tangent  line 


D,.  =  Ce-.1^(- ^)  =  4, 


m  =  VxV  =  C'6    2(r2| I  = ^xe    ^a^- 


Differentiating  a  second  time,  using  the  rule  for  the  deriva- 
tive of  a  product,  DxUv  =  u  DxV  +  v  DxU  (Exercise  21,  page  278), 
and  equating  the  result  to  zero,  we  have 

C/    —^  (     x\       _- 5i\ 
T>xm  = -^xe  2(rW ^ )  +  e  2<rO  =  0. 


Hence  5— 

Therefore  x  =  ^  cr. 


*  Since  any  number  is  the  logarithm  of  some  other  number  the  constant 
of  integration  may  be  written  in  the  form  loge  C.  This  is  convenient 
here  on  account  of  the  ease  with  which  log,  y  -  log«  C  may  be  transformed 

to  log.  |- 


408  ELEMENTARY   FUNCTIONS 

Hence  the  value  of  o"  gives  the  abscissas  of  the  points  of  in- 
flection.    The  ordinates  of    these  points  are  both  equal  to 

The  equation  of  the  probabiUty  curve  was  originally  de- 
rived in  connection  with  the  errors  arising  in  measurement. 
The  difference  between  a  measurement  and  the  unknown  true 
value  of  the  magnitude  measured  is  called  an  error,  and  this 
is  to  be  distinguished  from  the  deviation  of  one  of  a  set  of 
measurements  from  the  arithmetic  mean  of  the  set.  It  is 
assumed  that  as  the  number  of  measurements  increases  the 
mean  approaches  the  true  value,  and  the  deviations  from  the 
mean  become  more  nearly  the  true  errors. 

It  is  also  assumed  that  errors  are  distributed  symmetrically 
about  the  true  value  and  that  they  conform  to  the  laws  of 
probabiHty.     Hence  the  law  of  errors  is  given  by  equation  (1). 

From  this  point  of  view,  an  ordinate  yi  of  the  probability 
curve,  or  curve  of  errors,  gives  the  probability  of  an  error  equal 
to  the  corresponding  abscissa  Xi,  that  is,  the  probabiUty  of 
an  errot  Xi  is 

2/1  =  Ce    2a\ 

In  actual  measurements  there  is  a  lower  limit  to  the  size 
of  the  unit  employed  in  making  the  measurements.  The 
error  curve  can  be  thought  of  as  a  histogram  in  which  the  width 
of  a  column  is  Ax,  the  magnitude  of  the  unit  used,  and  its 
height,  y,  is  the  probability  that  an  error  will  fall  in  the  cor- 
responding class  interval.  If  all  the  columns  are  placed  on 
end  the  total  length  is  2?/  =  1,  since  this  is  the  sum  of  the 
probabilities  of  all  possible  errors.  (Theorem  1,  Section  136.) 
Since  the  area  of  the  histogram  is  proportional  to  the  sum  of 
the  corresponding  ordinates,  the  distribution  of  errors  may  be 
measured  by  means  of  the  areas  of  the  corresponding  columns. 
Hence  we  have  for  the  probability  curve: 

The  probability  that  an  error  will  be  between  two  values 
xi  and  X2  is  given  by  the  area  under  the  curve  between  these 
limits. 

Determination  of  the  constant  C.    Since  all  errors  will  be 


THEORY  OF  MEASUREMENT 


409 


included  between  the  limits  -  oo  and  +  cx),  the  area  between 
the  curve  and  its  asymptote  *  must  represent  certainty,  and 
hence  be  equal  to  unity.  It  can  be  shown  that  the  area  be- 
tween the  curve 

y  =  e  2<r2, 

and  its  asymptote  is  o'V27r,  a  result  which  we  assume.  Then 
the  area  between  the  curve 

y  =  Ce  2a2, 

and  its  asymptote  is  0^/21:  C,  since  the  latter  curve  may  be 
obtained  from  the  former  by  multiplying  the  ordinates  by  C. 

Hence  a  V27r  C  =  1. 

1 


Whence 


C  = 


(tV2w 
Therefore  the  equation  of  the  probability  curve 

1 


y  = 


o-\/27r 


(2) 


" 

n 

- 

Yy 

> 

"^ 

•^ 

y 

\ 

^ 

1 

_ 

\ 

/ 

1 

t 

Y 

27r 

1 

\ 

V 

y 

1 

'V 

s 

^ 

1 

■)  ■\ 

1 

V 

*s 

- 

— 

1 

<T 

<r 

1 

■"^ 

-1 

0 

1 

X 

_ 

Graph  of  y 


1 


<rV27r 
Fig.  222. 


1 


.56 


*  Let  A  denote  the  area  bounded  by  the  probability  curve,  the  a;-a3«s, 
and  the  ordinates  x  =  -  a  and  x  =  a.  It  can  be  shown  that  A  approaches 
a  definite  limit  as  a  becomes  infinite.  This  limit  is  called  the  area  between 
the  curve  and  its  ai  ymptote. 


410 


ELEMENTARY  FUNCTIONS 


In  the  theory  of  measurements  it  is  customary  to  set  h^ 
so  that  equation  (2)  becomes 

Vtt 


2(72 


(3) 


Example.  Find  the  equation  of  an  ideal  frequency  curve  (probability 
curve)  which  fits  approximately  the  upper  polygon  in  the  figure  at  the 
beginning  of  the  section.  Test  the  accuracy  with  which  the  curve  fits 
the  polygon  by  finding  the  values  of  y  given  by  the  equation  for  the  values 
of  X  given  in  the  table  and  the  amounts  by  which  these  values  differ  from 
the  given  values  oi  y. 

Changing  the  notation  by 
replacing  m  and  /  by  x  and  y 
respectively,  and  taking 
Ax  =  1,  the  table  at  the  be- 
ginning of  the  section  may 
be  given  as  in  the  first  two 
columns  of  the  acompanying 
table. 

The  standard  deviation  for 
this  table  is  c  =  1,  since  we 

saw  that  <7  =  Ax,  and  here  Ax  =  1.    The  equation  required  is  found  by 

substituting  this  value  of  o"  in  (2)  which  gives 


Given 

Computed 

Difference 

X 

y 

y 

-2 

tV  =  .063 

.054 

-  .009 

-1 

i  =  .250 

.242 

-  .008 

0 

1  =  .375 

.399 

+  .024 

1 

i=  .250 

.242 

-  .008 

2 

tV  =  .063 

.054 

-  .009 

2/  = 


1 
V2^ 


_l2 

e   2. 


Substitution  of  the  given  values  of  x  in  this  equation  gives  the  numbers 
in  the  third  column  of  the  table.  The  computation  requires  the  use  of  a 
table  of  values  of  the  exponential  function,  and  is  effected  by  the  use  of 
logarithms. 


EXERCISES 

1.  Construct  the  center  frequency  polygon  in  the  figure  at  the  beginning 
of  Section  141,  find  the  equation  of  an  ideal  frequency  curve  which  ap- 
proximates the  polygon,  and  plot  the  curve  on  the  same  axes  as  the  polygon. 
Find  the  difference  between  the  ordinate  of  each  vertex  of  the  polygon 
and  the  corresponding  ordinate  of  the  curve. 

2.  In  determining  the  constant  of  integration  in  equation  (1)  of  the 
preceding  section  the  area  between  the  curve  and  its  at^ymptote  is  some- 


Grade 

Frequency 

95 

8 

85 

22 

75 

24 

65 

17 

55 

6 

45 

3 

THEORY  OF  MEASUREMENT  411 

times  taken  equal  to  n,  the  sum  of  the  frequencies.    Show  that  in  this  case 
the  equation  becomes 

What  would  equation  (3)  become  in  this  case? 

3.  If  a  coin  is  tossed  eight  times,  find  the  frequencies  of  each  of  the 
various  ways  it  can  fall.  Plot  these  frequencies  as  ordinates  at  a  unit  dis- 
tance from  each  other,  and  draw  the  frequency  polygon.  Find  the  equation 
of  the  smoothed  frequency  curve  by  means  of  the  preceding  exercise. 

4.  Find  the  arithmetic  mean  and  the  standard  deviation  for  the  dis- 
tribution of  grades  in  trigonometry  given  in  the  table,  where  the  mid- 
values  of  the  class  intervals  for  the  grades  are  given. 
Plot  the  frequency  polygon.  Find  the  equation  of 
the  smoothed  frequency  curve  using  Exercise  2. 
Plot  the  curve  on  the  same  figure  as  the  polygon, 
using  as  origin  the  point  on  the  x-axis  which  repre- 
sents the  mean,  and  computing  the  values  of  y  for 
the  values  of  x  corresponding  to  the  mid-values  of 
the  class  intervals  given  in  the  table. 

5.  Find  the  arithmetic  mean  and  the  standard  deviation  of  the  heights 
of  12-year  old  boys  given  in  Exercise  6,  page  19.  Find  the  equation  of 
the  smoothed  frequency  curve,  and  plot  the  frequency  polygon  and  curve 
in  the  same  figure. 

6.  Plot  the  graphs  ofy  =  -^ — j-^  and  y  =  — -j.,  and  compare  them 

with  the  probability  curve.     Might  either  of  them  be  used  in  place  of  the 
probability  curve? 

142.  Probable  Error.  The  value  r  of  x  such  that  half  the 
errors  lie  between  the  limits  —  r  and  4-  ^  is  called  the  probable 
error. 

The  probable  error  is  a  measure  of  the  deviation  of  the 
measurements  from  the  true  value.  Since  in  a  symmetrical 
distribution  the  mean  and  median  coincide,  in  this  type  of 
distribution  the  probable  error  is  the  same  as  the  quartile 
deviation  (Qs  —  Qi)  /2.  Graphically  it  is  the  abscissa  on  the 
X-axis  whose  ordinate  bisects  the  area  of  one  branch  of  the 
curve. 

The  area  under  the  curve  y  =  —7=^  e     ^  from  -  hxto  +  hx  is 

VTT 

given  in  the  following  table  for  different  values  of  hx. 


412 


ELEMENTARY  FUNCTIONS 


hx 

A 

0 

0 

.1 

.112 

.2 

.223 

.3 

.329 

.4 

.428 

.5 

.521 

.6 

.604 

.7 

.678 

hx 

A 

hx 

A 

.8 

.742 

1.6 

.976 

.9 

.797 

1.7 

.984 

1.0 

.843 

1.8 

.989 

1.1 

.880 

1.9 

.993 

1.2 

.910 

2.0 

.995   . 

1.3 

.934 

2.1 

.997 

1.4 

.952 

2.2 

.998 

1.5 

.966 

2.3 

.999 

From  the  table  it  is  seen  that  the  area  is  A  =  |  when  kx  is 
between  0.4  and  0.5.     The  calculation  of  hr  to  four  places, 
which  is  beyond  the  scope  of  this  course,  gives 
hr  =  0.4769. 


Hence 


Therefore 


r  = 


0.4769 


0.4769(V2(7)  -  0.6745(7. 


0.6745.  &'. 


(1) 


The  probable  error  is  here  expressed  in  terms  of  the  unknown 
errors  x.  In  a  particular  case  only  the  deviations  from  the 
mean  can  be  calculated.  If  we  change  from  errors  to  devia- 
tions in  the  formula,  the  numerator  is  diminished,  since  Xd? 
<Xx^  (the  sum  of  the  squares  of  the  deviations  from  the  mean 
is  smaller  than  from  any  other  value).  It  is  assumed  in  practice 
that  the  best  value  of  the  denominator  to  conform  to  this 
change  is  n  -  1.  

Hence  r  =  0.6745. /^  (2) 

\n-  1 


and 


(3) 


If  two  sets  of  measurements  of  unequal  precision  are  com- 
pared it  will  be  found  that  while  the  areas 
under  the  probability  curves  are  the  same, 
viz.,  unity,  the  measurements  in  the  more 
precise  set  will  cluster  more  closely  about 
the  mean,  the  value  of  a  for  this  set  will  be 
smaller,  and  since  h  =  1  /a'V2,  the  value  of 


THEORY  OF  MEASUREMENT  413 

h  will  be  larger.  In  the  figure  the  curve  A  represents  the 
set  with  greater  precision. 

The  value  h  is  called  the  measure  of  precision  of  a  set  of 
measurements.  Since  the  intercept  on  the  ?/-axis  is  h  / Vtt,  this 
intercept  is  proportional  to  the  measure  of  precision. 

In  comparing  different  sets  of  measurements  the  weight  at- 
tached to  the  arithmetic  mean  of  any  set  (Theorem  3,  page  418) 
is  given  by  h^,  whose  value,  from  (3),  is 

Probable  Errors  in  Calculations 

Theorem  1.  If  y  =  f{x),  if  the  probable  error  in  x  is  r,  and  if 
the  probable  error  in  y  is  R,  then  R  =  D^y  r. 

Let  dxif  dx2f  •  .  •,  dxn  denote  the  errors  in  the  measure- 
ments of  x  and  dyi,  dy2,  .  .  .,  dyn,  the  corresponding  errors  in  y. 

Then  by  (1) 

R  =  0.6745  JH^       and       r  =  0.6745  JIM'. 

But  by  the  theorem  on  page  294, 

dyi  =  D^ydxi,  dy2  =D:,ydx2,  .  .  .,  dyn  =  B^ydXn. 
Squaring  each  of  these  equations  and  adding  the  results, 

^{dyY  =  lD,yJ^{dx)\ 
Extracting  the  square  root  of  both  sides  of  this  equation 
and  multiplying  both  sides  of  the  result  by  0.6745 /Vn, 

Hence  R  =  D^yr. 

Theorem  2.  //  y  is  a  function  of  several  variables,  y 
=  /(x,  2,  w)  and  if  the  probable  errors  in  x,  z,  w,  and  y  are  re- 
spectively ri,  r2,  rs,  and  R,  then 

R'  =  (D.yW  +  {D.yYr^^  +  {D^yfr^K 

Jj^tdxijdxiy  .  .  .,  dxn,dz\,dz2,  .  .  .,  dzn,dwi,dw2,  .  .  .,  dwn 


414  ELEMENTARY  FUNCTIONS 

be  the  errors  in  x,  z,  and  w  for  n  sets  of  measurements  and  dyi, 
dy2,  .  .  .,  dyn  the  errors  in  y  corresponding  to  each  set  of 
measurements,  respectively. 

Then      *dyi  =  {D^y)dxi  +  {D^y)dzi  +  {Dy,y)dwi 


dyn  =  {Dxy)dxn  +  {Dzy)dZn  +  {Dy,y)dwn. 
Squaring  and  adding  these  equations,  we  have 
S(d2/)2  =  (Z).2/)lS(da;)2]  +  {D.yYl^idzy]  +  {D^y)-^l^{dwn. 

The  product  terms  are  omitted,  because  in  the  long  run  there 
will  be  as  many  positive  products  as  negative,  and  as  they  will 
be  distributed  according  to  the  law  of  error,  they  will  cancel 
each  other.  The  squared  terms  are  all  positive  and  hence 
cannot  cancel. 

Multiplying  both  sides  of  the  last  equation  by  (0.6745) V^, 
it  follows  that 

m  =  {D,y)W  +  {B:,y)W  +  (D.2/)W. 

Example.  Find  the  probable  error  R  in  the  value  of  the  acceleration 
of  gravity  g  as  determined  by  a  seconds  pendulum,  if  five  measurements 
of  the  time  of  vibration  t,  in  seconds,  are  1.0028,  1.0006,  0.9994,  0.9990, 
0.9982,  and  five  measurements  of  the  length  I,  in  centimeters,  are  100.18, 
100.12,  99.90,  99.96,  99.84,  given  that 

9  =  -^' 

*  The  following  extension  of  the  theorem  on  small  errors  on  page  294 
is  proved  in  works  on  the  calculus: 

If  y  is  a  function  of  several  independent  variables  x,  z,  w,  then  an  ap- 
proximate value  of  the  error  in  y  due  to  errors  of  dx,  dz,  dw,  in  the  inde- 
pendent variables  is  given  by 

dy  =  Dxy  dx  +  Dzy  dz  +  D^y  dw. 
In  obtaining  Dry,  z  and  w  are  regarded  as  constants.     A  different  no- 
tation for  the  derivative  of  y  with  respect  to  x  alone,  namely,  -r^,  is  usually 

ox 

used  when  t/  is  a  function  of  other  variables  besides  x,  but  it  seems  hardly 
worth  while  to  introduce  it  in  this  connection. 


THEORY  OF  MEASUREMENT 


415 


The  means  of  t  and  I  and  their  probable  errors  are  calculated  as  follows: 


t 

d 

d^ 

1.0028 

+  28x10-^ 

784  X  10-« 

1.0006 

+    6x10-^ 

36  X  10-8 

.9994 

-    6x10-4 

36xl0-« 

.9990 

-lOxlO--* 

100xlO-« 

.9982 

-18x10-" 

324  X  10-8 

At 

=  1.0000 

2d2 

=  1280x10-8 

I 

d 

(i2 

100.18 

+  18x10-2 

324  X  10-* 

99.90 

-  10  X  10-2 

100  X  10-« 

99.96 

-    4x10-2 

16  X  10-* 

99.84 

-  16  X  10-2 

256  X  10-* 

100.12 

+  12  X  10-2 

144  X  10-* 

A I  =  100.00 

2d2  =  840x10-* 

By  (2)  we  have: 

Probable  error  tj  =«  0.6745 


Probable  error  n  =  0.6745 
Differentiating  the  given  equation, 


v' 


1280  X  10-8 


=  0.0012  seo 


840  X  10-" 


=  0.10  cm. 


Big  =  -^, 


27r2Z 


Substituting  the  mean  values  of  t  and  I, 


(Theorem  2) 


(0.0012)* 


=  0.974  +  5.61 
/.  R  =  2.6. 
The  value  of  ^  is 


6.584 


g  = 


irH     irnOO 


=  987.0. 


Hence 


t^  1 

g  =  987.0  =t  2.6 

EXERCISES 

1.  Find  the  probable  error  in  the  circumference  of  a  circle  whose  diam- 
eter is  2.14  centimeters  with  a  probable  error  of  ±  0.08. 

2.  The  length  and  breadth  of  a  rectangle  were  measured  in  centimeters 
and  the  following  values  obtained: 


Length 


54.32, 


54.35, 


54.36, 


54.31, 


54.33 


Width  25.64,  25.62,  25.63,  25.61,  25.65 

Find  the  area  and  the  probable  error  of  the  area. 

3.  The  length  and  diameter  of  a  cylinder  were  measured  in  inches  and 
the  following  values  obtained. 
Length 


2.745, 

2.747, 

2.742, 

2.744, 

2.743 

0.872, 

0.870, 

0.873, 

0.873, 

0.874 

Diameter 
Find  the  volume  and  the  probable  error  in  the  volume. 


416  ELEMENTARY  FUNCTIONS 

4.  A.  measurement  is  recorded  as  7.50  with  a  probable  error  of  r  =  ±  0.32. 
What  is  the  probability  that  an  error  will  be  less  than  0.16? 

Suggestion.  Since  h  =  0.4769/r,  hx  =  0.4769a; /r  =  0.4769  X  0.16/0.32 
=  0.24.  Use  the  table  of  areas  to  find  the  probability.  What  is  the  prob- 
ability that  the  error  will  be  less  than  0.64?  greater  than  0.64? 

-  5.  A  line  is  measured  25  times  and  the  value  of  the  measure  of  preci- 
sion found  to  be  0.2  inch.  How  many  errors  are  less  than  1  inch?  greater 
than  1  but  less  than  2  inches?  greater  than  2  inches? 

6.  What  is  the  probable  error  of  the  sum  or  difference  of  two  magni- 
tudes whose  probable  errors  are  n  and  r2?  of  the  product  of  the  two  magni- 
tudes? of  the  quotient? 

7.  If  the  sides  a  and  6  of  a  right  triangle  are  34.26  ±  0.14  and  77.81  =«=  0.23 
what  is  the  probable  error  in  tan  A? 

8.  A  length  of  1000  feet  is  measured  with  a  steel  tape  100  feet  in 
length.  What  is  the  probable  error  in  the  length  if  the  probable  error  in 
the  tape  is  0.01  foot? 

9.  What  would  be  the  probable  error  of  an  observation  if  the  law  of 
the  distribution  of  errors  were 

(a)  y  =  b,  from  x  =  -  o  to  x  =  +  a?  In  a  seven-place  table  of  loga- 
rithms the  seventh  place  is  never  in  error  by  more  than  0.5;  what  is  the 
probable  error? 

(b)  y  =  a  -  X  from  x  =  0  to  x  =  a  and  y  =  a -\- x  from  x  =  -  a  to  x  =  0? 

(c)  y  =  -  x"^  +  a^  from  x  =  -atox  =  +a? 

143.  Least  Squares.  Let  z  represent  the  most  probable 
value  of  a  set  of  n  measurements  Xi,  X2,  .  .  .,  Xn,  which  are 
supposed  equally  precise.    Then  the  deviations  from  z,  namely 

Z  —  Xi,    Z  —  X2,    .    .    .,    Z  —  Xn 

will  be  distributed  according  to  the  law  of  probability. 

The   probabilities   of   the   several   deviations   are    (by   (3), 

page  410) 

h     -f^\'-^y 

Vtt 


Vtt 


h     -^^(--«)' 


h     -^\'-"^y 
Vtt 


As  the  measurements  are  independent,  the  probability  of 
this  particular  system  is  (Theorem  2,  page  376) 


THEORY  OF  MEASUREMENT  417 

^  =  ^r^^  *  (1) 

The  probability  P  will  have  a  maximum  value  when  the 
second  factor  in  the  exponent, 

(Z  -  XiY  +  (2  -  X^Y  +    ...    (2  -  XnY,  (2) 

has  a  minimum  value.    Hence  we  have  the 

Principle  of  least  squares.  The  most  probable  value  of  an 
observed  magnitude  which  has  been  repeatedly  measured  gives 
the  least  possible  value  to  the  sum  of  the  squares  of  the  deviations 
of  the  measurements. 

To  find  the  value  of  z  for  which  the  sum  of  the  squares  above 
is  a  minimum,  we  differentiate  with  respect  to  z  and  equate 
the  derivative  to  zero.    This  gives 

2(2  -  Xi)  +2{Z-X2)  +    .    .    .    +  2(2  -  Xn)   =  0. 

Hence  nz  =  xi-\-X2+  .  .  .  +Xn 

J  Xi  +  X2  +    .    .    .    +  Xn  ,o\ 

and  z  =  .  (3) 

n  ^  ^ 

Hence  we  have 

Theorem  1.  The  best  valve  to  represent  a  set  of  equally  precise 
measurements  is  their  arithmetic  mean. 

The  probable  error,  R,  of  the  arithmetic  mean,  z,  is  found 
by  Theorem  2  of  the  preceding  section  to  be 

where  the  r's  are  the  probable  errors  of  the  measurements. 
As  the  measurements  are  supposed  equally  precise,  let  r  be 
the  common  value  of 

fi  =  r2  =  r3  =  .  .  .   =  r„. 

Hence  R^  =  n(^%\ 

r 
so  that  R  =  -7='  (4) 

Vn 

Hence  we  have 

Theorem  2.     The  probable  error  of  the  arithmetic  mean  of  a 

set  of  eqvxdly  precise  measurements  varies  inversely  as  the  sqvxire 


418 


ELEMENTARY  FUNCTIONS 


root  of  the  number  of  measurements.  The  probable  error  de- 
creases as  the  number  of  measurements  increases,  but  less  rapidly. 

Theorem  3.  The  most  probable  valu£,  z,  for  a  set  of  arithmetic 
means  Xi,  X2,  .  .  .  Xn,  of  unequal  precision  hi,  hz,  .  .  ,  hn,  is 
the  arithmetic  mean  of  the  means  with  the  weights  h^,  hi,  .  .  .  h^. 

That  is 

hi^Xi  +  h2^X2  +    .    .    .    +  hn^Xn 
h'  +  h2'+    .    .    .    +  hn' 

The  proof,  which  is  left  as  an  exercise,  is  analogous  to  that 
of  Theorem  1.  In  applying  equation  (3),  page  410,  the  value 
of  h  is  different  for  the  different  measurements. 

In  an  empirical  data  problem,  the  most  probable  values  of  the 
constants  in  the  desired  equation  may  be  calculated  by  the  method 
of  least  squares. 

Example.  Find  the  most  probable  values  of  the  coefficients  of  a 
linear  function  which  will  represent  the  following  empirical  table  of  values. 


Sd2  =  (m  +  6- 1.3)2 +  (2m  +  &- 1.6)2  + (3m  +  6 -3.1)2 +  (4m  +  6 -2.8)2.  (i) 
Differentiating  the  sum  of  the  squares  of  the  deviations  with  respect  to 
m  and  equating  the  result  to  zero,  we  have 

2(w  +  fe-1.3)+2(2m  +  fe-1.6)2  +  2(3m  +  &-3.1)3  +  2(4m  +  6-2.8)4  =  0 
or       (1  +  4  +  9  +  16)m  +  (1  +  2  +  3  +  4)6  =  1.3  4-  3.2  +  9.3  +  11.2. 
Hence  30m  +  106  =  25.  (2) 

Differentiating  (1)  with  respect  to  h  and  equating  the  result  to  zero  we  have 
2(m  +  6  -  1 .3)  +  2(2m  +  6-1.6)+  2(3w  +  6-3.1)+  2(4m  +  6  -  2. 8)  =  0 
or  (1  +  2  +  3  +  4)m  +  46  =  1.3  +  1.6  +  3.1  +  2.8. 

Hence  lOw  +  46  =  8.8  (3) 

Solving  equations  (2)  and  (3)  simultaneously,  we  have 

m  =  0.6        and        6  =  0.7. 
Hence  the  required  function  is    2/  =  0.6a;  +  0.7. 
By  the  method  of  Section  27,  y  =  0.68x  +  0.50. 


Empirical 

y 

Theoretical 

y 

Deviation  of  theoretical  y 
from  empirical  y 

1.3 
1.6 
3.1 

2.8 

w  +  6 
2w  +  6 
3w  +  6 
4w  +  6 

m  +  6-  1.3 
2m +  6 -1.6 
3m  +  6-3.1 
4m  +  6-2.8 

THEORY  OF  MEASUREMENT  419 

For  this  method  the  errors  are  -  0.12,  +  0.26,     -  0.56,     +  0.42 

or  -9%,    +16%,     -18%,    +15% 

and  the  standard  deviation  of  the  errors  is  o-  =  0.378.     For  the  method  of 

least  squares  the  errors  are 

0,  +  0.3,    -  0.6,     +  0.3 
or  0  %,  +  20%,  -  19  %,  +  11  % 

with  a  standard  deviation  of  the  errors  of  o"  =  0.368. 

EXERCISES 

1.  If  the  pairs  of  values  {xi,  yi),  (X2,  2/2),  ..  .  (Xn,  yn)  of  an  empirical 
table  are  connected  by  the  relation  y  =  mx  show  by  the  method  of  least 
squares  that  the  most  probable  value  of  m  is  m  =  Sxy/Sx^. 

2.  In  a  psychological  experiment  to  determine  the  ability  of  an  in- 
dividual to  determine  variation  in  pressure,  the  following  table  was  ob- 

Initial  weight  in  grams  I     10,      20,    30,     40,      t^^®^-       Find 

the    law,     and 

Just  perceptible  weight  in  grams  •  5.15,  7.8,  9.9,  13.3,    ^^le   change   in 

weight  which  is  just  perceptible  when  the  initial  weight  is  50  grams. 

3.  A  steel  bar  was  stretched  by  attaching  weights,  and  the  measure- 


Tension  in  pounds 


150,  400,  500,    700,     ^^^^^  ^^^^  ^  g^^en  in  the 
table.      Determine  the  law, 


Stretching  in  inches      .32,  .80,  .98,  1.44,    ^^^  ^^  y^^^  ^^^^  ^^^  b^^ 

would  be  stretched  by  a  weight  of  800  pounds. 

4.  The  table  gives  the  horse  power  required  for  a  given  load  in  the 
Load  in  tons     50,    70,     90,    100,     case  of  a  locomotive  running  40  miles 

an   hour.     Determine  the   law,  and 


81,  107,  127,  142,     gjj(j  ^jjg  horse  power  required  for  a 


Horse  power 
load  of  68  tons. 

5.  If  the  values  (xi,  yO,  (xj,  t/2),  .  .  .  (x«,  yn)  of  an  empirical  table 
are  connected  by  the  relation  y  ==  mx  ■{•b,  show  that  the  most  probable 
values  of  m  and  b  are  given  by  the  equations 

_ni:(xy)  -S(a;)S(t/)  ^  _  S(y)  -  mi:(x) 

nX(x^)  -  (2:(x))2  '  n 

6.  Use  the  results  of  the  preceding  exercise  to  calculate  the  coefl&cients 
of  the  linear  function  P  =  mW  +  6  in  Example  2,  Section  27,  arranging 
the  work  with  columns  headed  x,  y,  x^,  xy.  Compare  the  results  with  those 
obtained  in  Section  27. 

7.  If  the  values  (xi,  yx),  (xz,  2/2)  ..  .  {Xn,  yn)  of  an  empirical  table  are 
connected  by  a  relation  of  the  form  y  =  ax^  +  6x  +  c  show  that  the  most 
probable  values  of  the  coefficients  a,  b,  c,  are  given  by  the  equations 

aS(x4)  +  6S(x3)  +  c2(x2)  =^{x^y), 
a2(x3)  +  52(x2)  +  cS(x)  =  S(X2/), 
aS(x2)+bS(x)  -\-cn        =2(2/). 

8.  Use  the  results  of  the  preceding  exercise  to  calculate  the  most  prob- 
able values  of  the  coefficients  of  the  quadratic  function  representing  the 


420  ELEMENTARY  FUNCTIONS 

empirical  table  in  the  example  in  Section  35.  Arrange  the  work  with 
columns  headed  x,  y,  xy,  x^,  a^,  x'^y,  x^.  Compare  the  results  with  the 
values  obtained  for  the  coefficients  by  the  method  of  Section  35. 

9.  If  the  values  (xi,  yi),  {Xi,  2/2)  ..  .  (Xn,  yn)  of  an  empirical  table  are 
connected  by  a  relation  of  the  form  y  =  fcx"»  find  the  most  probable  values 
of  the  coefficients,  given  that  Dx  loge  u  =  Dxu/u.  If  the  relation  is  of  the 
form  y  =  k^"  find  the  most  probable  values  of  the  coefficients. 

10.  Determine  the  most  probable  values  of  the  constants  in  an  equa- 
tion of  the  form  y  =  kx""  connecting  the  following  values  of  x  and  y, 
(5,  11),  (10,  31),  (15,  59),  (20,  88).  Suggestion:  Find  log  x  and  log  y, 
rounded  off  to  two  figures,  and  then  proceed  as  for  a  linear  function. 

11.  The  temperature  of  a  body  cooling  in  air  at  zero  temperature  was 
78°,  60°,  48°,  36°  at  the  end  of  1,  2,  3,  4,  minute  intervals  respectively. 
Determine  the  most  probable  values  of  the  rate  of  cooling  and  the  original 
temperature. 

12.  Six  measurements  of  a  line  with  a  steel  tape  were  in  feet  639.15 
639.08,  639.11,  639,19,  639.10,  639.14,  and  with  a  chain  639.2,  639.1, 
639.1,  639.4,  639.3,  639.0.  Find  the  means  of  the  two  sets  of  measure- 
ments and  their  weights  and  the  weighted  mean. 

13.  Find  the  most  probable  value  of  the  velocity  of  light  from  the  fol- 
lowing determinations  in  kilometers  per  second:  298,000  =tlOOO  ;  298,500  ± 
1000;  299,990  ±  200;  300,100  =*=  1000;  299,930  ±  100. 

14.  A  line  is  measured  10  times  and  the  probable  error  of  the  mean  is 
0.012  foot.  How  many  additional  measurements  of  the  same  precision 
are  required  to  reduce  the  probable  error  of  the  mean  to  0.005  foot? 

15.  The  parallax  of  the  sun  by  two  determinations  is  8.883  ±  0.034  and 
8.943  =t  0.051.  What  are  the  weights  of  the  two  observations?  What  is 
their  weighted  mean? 

144.  Correlation.  In  the  experiment  to  determine  the  co-  ^ 
efficient  of  friction  of  wood  on  wood,  discussed  in  Section  27, 
the  causal  relation  between  the  two  weights  is  apparent.  To 
a  change  in  one  weight  corresponds  a  change  in  the  other.  An 
arbitrary  value  can  be  given  to  either  weight  and  the  cor- 
responding value  of  the  other  determined,  and  the  pairs  of 
values  found  by  the  two  methods  are  the  same.  That  is, 
the  direct  and  inverse  relations  are  given  by  the  same  equation. 
In  short,  one  variable  is  a  function  of  the  other. 

When  some  variables  are  compared,  for  example,  the  price 
of  roses  and  the  amount  of  salt  mined,  no  causal  relation  can 
be  detected. 

Between  the  extremes  of  two  variables  which  are  causal 
dependents  or  absolutely  unrelated,  there  are  varying  degrees 


THEORY  OF  MEASUREMENT 


421 


of  dependence.  In  this  section  we  shall  consider  the  kind  of 
relation  given  in  the 

Definition.  Two  variables  are  said  to  be  correlated  if  an 
increase  in  one  is  accompanied  either  by  an  increase  or  by  a 
decrease  in  the  other.  A  measuri  of  the  degree  of  dependence 
is  called  a  coefficient  of  correlation. 

The  coefficient  of  correlation  measures  the  probability  with 
which  the  value  of  one  variable  may  be  predicted  from  an 
assumed  value  of  the  other. 

The  correlation  is  said  to  be  perfect  if  a  causal  relation  is 
established  between  the  two  variables,  that  is,  if  one  is  a  func- 
tion of  the  other.  The  correlation  is  said  to  be  positive  or 
negativCf  according  as  an  increase  in  one  variable  is  accom- 
panied by  an  increase  or  decrease  in  the  other. 

The  values  of  two  associated  variables  may  be  conveniently 
arranged  in  rows  and  columns.  Such  an  array  is  called  a 
correlation  table. 

Example  1.  Represent  the  following  correlation  table 
graphically. 

Correlation  between  July  precipitation  and  yield  of  corn  in 
Ohio  for  the  years  1854-1913  inclusive. 


Mid-values  of 

Precipitation  in  inches 

fy 

class  intervals 

1.5 

2.5 

3.5 

4.5 

5.5 

6.5 

7.5 

8.5 

g 

42.5 

1 

1 

2 

1 

1 

6 

-    a; 

37.5 

1 

7 

8 

7 

1 

. . . 

24 

^1 

32.5 

1 

5 

8 

4 

1 

1 

. . . 

20 

-f    S 

27.5 

1 

5 

1 

2 

9 

1^ 

22.5 

1 

1 

^ 

/x 

'2 

12 

16 

16 

10 

3 

0 

1 

60 

Each  row  is  a  frequency  distribution  for  the  class  interval 
of  y  given  at  the  left,  and  each  column  for  the  class  interval 
of  X  given  at  the  top. 

The  class  interval  for  the  precipitation  is  one  inch,  and  for 
the  corn  yield  five  bushels  per  acre. 


422 


ELEMENTARY  FUNCTIONS 


The  number  8  in  the  row  and  column  headed  respectively 
37.5  and  4.5  means  that  in  8  different  years  a  yield  of  35  to  40 
bushels  of  corn  per  acre  was  associated  with  a  precipitation  of  4  to 
5  inches  of  rain.  The  other  frequencies  have  similar  meanings. 
The  column  on  the  right  and  the  row  at  the  bottom  of  the 
table  give  respectively  the  sums  of  the  frequencies  for  each 
class  interval  of  y  and  x.  The  sum  of  the  right-hand  column, 
or  of  the  bottom  row,  is  the  total  number  of  items,  n  =  60. 

A  graphical  representation  of  the  table  is  obtained  by  plotting 
the  tables  below.    Table  1  gives  the  mid- values  of  the  x  classes 

and  the  mean  values  of  the  cor- 
Tablel  Table  2  responding     vertical     columns. 

The  pairs  of  values  in  this 
table  are  plotted  as  small  circles 
in  Fig  224.  In  this  instance,  as 
in  many  cases,  the  circles  lie  ap- 
proximately on  a  straight  line  Zi. 
Table  2  gives  the  mid- values 
of  the  y  classes  and  the  mean 
values  of  the  corresponding  hori- 
zontal rows.  The  pairs  of  values 
are  plotted  as  small  squares,  and 
a  straight  line  h  may  be  fitted  to  them  approximately.  In  this 
case  the  small  square  at  the  point  (22.5,  4.5)  represents  but  one 
of  n  =  60  items,  and  may  be  neglected  in  drawing  I2. 

The  table  is  said  to  be  repre- 
sented by  these  lines.  As  they 
do  not  coincide,  the  direct  and 
inverse  relations  between  the 
July  precipitation  and  the  yield 
of  corn  are  not  given  by  the 
same  equation.  That  is,  neither 
of  these  variables  is  a  function 
of  the  other  variable  only. 

As  these  lines  have  positive 
slopes,  to  an  increase  in  either  variable  corresponds  an  increase 
in  the  other.    Hence  the  correlation  is  positive. 


X 

Mean  y 

1.5 

30 

2.5 

31.7 

3.5 

34.4 

4.5 

34.4 

5.5 

38 

6.5 

37.5 

8.5 

42.5 

7 

244.5 

Table  2 

y 

Mean  x 

22.5 

4.5 

27.5 

2.9 

32.5 

3.6 

37.5 

4.5 

42.5 

5.5 

5 

21.0 

M. 


My  =  34.9 


Y 

' 

— " 

"■ 

"~ 

— 

"— 

. 

■— " 

~" 

^ 

^- 

h 

/ 

f^ 

^ 

/ 

-.-» 

~ 

35^ 

V 

- 

■- 

-- 

- 

^ 

^ 

** 

^ 

f 

- 

^ 

-H 

/ 

— 

A 

- 

/ 

/ 

- 

h 

0 

: 

2 

c 

i 

5 

6 

3 

^ 

X 

- 

-L 

± 

lL 

- 

± 

Fig.  224. 


THEORY  OF  MEASUREMENT 


423 


The  mean  values  of  x  and  y  for  the  entire  correlation  table 
are  given  at  the  ends  of  tables  1  and  2.  These  values  may  be 
obtained  also  as  the  means  of  the  row  and  column  headed  /» 
and  jy.  In  practice,  they  are  computed  in  connection  with  other 
quantities,  as  in  Example  3  below.  Notice  that  the  lines  h  and 
h  appear  to  intersect  at  the  point  M{Mx,  My). 

As  in  Example  1,  the  direct  and  inverse  relations  between  two 
correlated  variables  are  usually  represented  by  two  distinct 
lines.  These  lines  coincide  if  the  correlation  is  perfect,  while 
li  and  h  are  parallel  to  the  x  and  ?/-axes  respectively  if  the 
variables  are  unrelated.  The  correlation  is  positive  or  negative 
according  as  the  slopes  of  these  Unes  are  positive  or  negative. 

The  line  h  representing  the  means  of  the  columns  always 
makes  a  smaller  angle  with  the  x-axis  than  the  line  h  repre- 
senting the  means  of  the  rows.  If  x  is  the  height  of  fathers 
and  y  that  of  all  their  sons,  the  sons  of  men  who  are  taller  or 
shorter  than  the  average  tend  to  approach  the  average  height 
more  closely  than  their  fathers.  On  this  account  Galton 
called  the  falling  back  of  the  lines  toward  the  axes  regression, 
and  hence  the  lines  are  generally  called  lines  of  regression. 

We  shall  assimie  the  fact  that  the  lines  of  regression  pass 
through  the  point  M  whose  coordinates  Mx  and  My  are  the 
mean  values  of  the  x  and  y  distributions  (see  Example  1). 
Let  x'  and  y'  be  new  variables  referred  to  axes  through  M 
parallel  to  the  old  axes.     Let  the  equations  of  ^i  and  h  be 

y'  =  mix', 
where  mi  is  the  slope  of  h  referred  to 
the  X-axis,  and 


x'  =  m22/', 
where  W2  is  the  slope  of  hz  referred  to  the 
y-axis.* 

The  most  probable  values  of  mi  and 
1712  can  be  determined  by  the  principle 
of  least  squares. 


(o,xj 


*  If  Pi  and  Pa  are  two  points  on  k,  and  if  02  is  the  inclination  of  k,  then 

X'l  -  x\ 


m%  = 


y'x  -  y'i 


cot  Q%, 


424  ELEMENTARY  FUNCTIONS 

For  a  given  value,  x\,  the  deviation  of  the  mean  value  of  the 
corresponding  column,  y\,  from  the  ordinate  of  the  point  on 
li  with  the  same  abscissa,  mix\,  is  mix\  -  y\.  The  sum  of  the 
squares  of  such  deviations  is 

^imix'  -  y'Y  =  S(mi2x'2  -  2mix'y'  +  y'^) 
=  mi2Sa:'2  -  2miSx'i/'  +  Zi/'^. 

Differentiating  with  respect  to  mi  and  equating  the  deriva- 
tive to  zero,  we  get 

2mi2x'2  -  2i:,x'y'  =  0. 

The  variables  x'  and  y'  are  nothing  but  deviations  from  Mx 
and  My,  and  the  particular  values  of  x  and  y  used  in  these 
summations  may  be  suggestively  denoted  by  dx  and  dy.  With 
this  change  in  notation,  Sx'y'  =  Sdx  dy]  and  Sa;'^  =  Srfj.2. 

XT  2x'w'      Sc?x  (ij/ 

Hence  '"i  =  S?^  =  Sd^' 

But  the  standard  deviation  of  the  dxB  is  given  by  (t^  =  — —> 

Til 

so  that  Zrfx^  =  naj^. 
Hence  the  slope  of  li  is 

mi  =  ^,        where        o-^^  = •  (1) 

Similarly,  the  slope  of  h  referred  to  the  y-Sixis  is 

m.  =  ^^        where        <r„»  =  ^-      '         (2) 
n(Ty  n 

The  slopes  mi  and  m2  are  respectively  measures  of  the  change 
of  y  with  respect  to  x  and  of  x  with  respect  to  y.  Their  geo- 
metric mean,  r  =  V  mim2,  is  usiuilly  chosen  as  the  coefficient  of 
correlation  to  represent  these  two  measures  of  the  direct  and  in- 
verse relations  between  the  variables.  Hence  the  value  of  the 
coefficient  of  correlation  used  almost  universally  is 

r  =  vWi;  =  ^^-  (3) 

n(Ty(Tx 

Substituting  the  value  of  l^dx  dy  obtained  from  (3)  in  (1)  and 
(2)  we  obtain 

mi  =  r^         and        m2  =  r-''  (4) 


THEORY  OF  MEASUREMENT  425 

If  ^1  and  I2  coincide,  and  if  6  is  their  inclination,  then 
r  =  Vmim2  =  Vtan  d  cot  6=^1. 

If  li  and  U  coincide  with  the  x'  and  ^/'-axes  respectively, 
then  r  =  Vmm2  =  Vo-O  =  0. 

Hence  for  perfect  correlation  r  =  =*=  1,  and  for  two  unrelated 
variables  r  =  0.  Values  of  r  between  +  1  and  —  1  indicate 
various  degrees  of  positive  and  negative  correlation. 

Example  2.  Find  the  coefficient  of  correlation  for  the  table  in  Ex- 
ample 1,  graphically. 

Reading  from  Fig.  224  the  coordinates  of  two  points  on  k,  and  dividing 
the  difference  of  the  ordinates  by  the  difference  of  the  abscissas,  we  find 

40-35      5      ^- 
mi  -  -j-;;^  -  3  =  1.7. 

Reading  from  the  figure  the  coordinates  of  two  points  on  h,  and  dividing 
the  difference  of  the  abscissas  by  the  difference  of  the  ordinates,  we  get 

5-4       1      no 


Hence  r  =  Vmimi  =  V1.7  x  0.2  =  \/0.34  =  0.58  approximately. 

The  calculation  of  the  coefficient  of  correlation  is  simplified 
by  the  following  considerations,  which  are  analogous  to  those 
used  in  computing  an  arithmetic  mean  (Theorem,  page  389), 
or  a  standard  deviation  (Theorem,  page  401). 

The  notation  employed,  some  of  which  has  been  used  above, 
is  as  follows: 

Mx  and  My  denote  the  true  means  of  the  variables. 

dx  and  dy  denote  deviations  from  the  true  means. 

(Tx  and  CTy  denote  the  standard  deviations  of  the  deviations 
from  the  true  means. 
— I 1 1-»        Ex  and  Ey  denote  estimated  values  of  the 


'y 

true  means. 


p,jQ  226  ^'  ^^^  ^'^  denote    deviations   from   the 

estimated  means. 

Cx  and  Cy  denote  the  corrections  used  in    calculating  the 

arithmetic  means,  such  that  Mx  =  Ex  +  Cx  and  My  =  Ey  +  Cy, 

The  two  sets  of  deviations  are  connected  by  the  relations 

a  X  ^  O'x  "T"  Cx 

and  d'y  =  dy  +  Cy, 


426 
Hence 


ELEMENTARY  FUNCTIONS 


=     2dx  dy    +    CxS   dy    +    CyE   dx    +    l^Cafiy. 

But  2c?x  =  Sdy  =  0,  since  the  sum  of  the  deviations  from  the 
mean  is  zero,  and  Sc^Cj,  =  nCxC^,  since  Cx  and  Cy  are  constants. 
Therefore 

Hid'jjd^  =  '^dxdy  +  nCxCy, 


or 


Xdxdy  =  2d'x  f^' 


'itf/- 


(5) 


This  equation  is  useful  in  calculating  the  coefficient  of  cor- 
relation r. 

The  probable  error  in  r  is  given  by  the  formula,  which  we 
assume, 

P.Er  =  0.67449^-^.  (6) 

Vn 

The  correlation  between  two  variables  is  usually  not  con- 
sidered as  established  unless  r  is  at  least  three  times  as  great 
as  the  probable  error. 

Example  3.  Calculate  r,  Mx,  My,  mi,  and  m2  for  the  table  in  Ex- 
ample 1. 

Let  the  class  intervals  be  chosen  as  units.  Assume  that  E^  =  4.5  and 
Ey  =  32.5.    The  work  can  be  conveniently  arranged  as  follows: 


Precipitation  in  inches 

fv 
6 

d'y 

fv^'v 

S^'\ 

12 

42.5 
37.5 

L5 

2.5 

1 
1 

3.5 

7 
8 
1 

4.5 

1 

8 
4 
2 

1 

5.5 

2 

7 
1 

6.5 

1 
1 
1 

7.5 

8.5 
1 

1 

2 

12 

24 

24 
20 

1 

24 

24 

0 

0 

14 

0 

32.5 

27.5 
22.5 

1 
1 

5 
5 

0 

0 

0 

n 

9 
1 

-  1 
-2 

-9 
-2 

9 

— 

— 

— 

— 

4 

^^^ 

^__ 

/x 

2 

12 

16 

16 

10 

3 

0 

1 

60 

d'x 

-3 

-2 

-  1 

0 

1 
10 
10 

2 

6 

12 

3 
0 
0 
0 

4 

4 

16 

8 

n  =S/x  =  2A  =  60. 

Ud'. 

-6- 

-24 

-  16 

0 

S/xd'x  =  -26.     S/Xy  =  25. 

f.d\ 

18 

48 

16 

0 

S/xd\=120.    2/^^  =  61. 

d':4'y 

3 

4 

-6 

0 

11 

6 

lld'^'y  =  26. 

THEORY  OF  MEASUREMENT  427 

The  frequencies  jx  are  the  sums  of  the  columns,  and  the  frequencies 
/„  are  the  sums  of  the  rows. 

The  deviations  d'x  are  deviations  from  E^  =  4.5  in  class  intervals  of  one 
inch.  The  deviations  d'y  are  deviations  from  E,,  =  32.5  in  class  intervals 
of  5  bushels  per  acre.  These  deviations  are  analogous  to  those  used  in 
Example  1,  page  390. 

The  products  fxd'x,  fxd'^x,  fyd'y,  fyd\  are  obtained  readily,  and  so  also 
are  the  sums  of  the  products. 

The  calculation  of  the  sum  Sd'xd'„  requires  elucidation.  There  are 
n  =  60  items  in  this  sum,  but  they  are  not  all  distinct  as  several  items  may 
have  the  same  deviations  d'x  and  d'y.  For  example,  the  precipitation 
was  2.5  inches,  and  the  yield  27.5  bushels  in  5  different  years.  There 
are  therefore  5  items  with  deviations  d'x  =  -2  and  d'y  =  -  1.  The 
sum  of  the  products  d'xd'y  for  these  5  items  is  therefore  5  (-2)  (-1)  =  10. 
Hence  the  sum  'Ld'xdy  may  be  obtained  by  multiplying  each  frequency 
in  the  body  of  the  given  table  by  the  product  of  the  values  of  d'x  and  d'y 
for  the  column  and  row  in  which  the  frequency  occurs,  and  then  adding 
these  products. 

A  more  condensed  method  is  worked  out  in  the  table.  The  column  on 
the  extreme  right  gives  the  partial  sums  of  the  products  just  described 
for  each  of  the  values  of  d'y.  To  obtain  the  first  partial  sum,  multiply 
each  frequency  in  the  row  headed  42.5  by  the  corresponding  value  of  d'xj 
add  these  products,  and  multiply  the  sum  by  d'y  =  2.  This  gives 
[l(-2)  +  l-0  +  2.1  +  l-2  +  l-4]x2  =  12. 

The  other  partial  sums  in  the  column  erroneously  but  conveniently 
headed  d'xd'y  are  obtained  in  like  manner.  The  sum  of  the  partial  sums 
in  this  column  is  ^d'jjd'y  =  26. 

This  procedure  may  be  varied  by  interchanging  the  words  row  and 
column,  d'x  and  d'y.  The  bottom  row  in  the  table  gives  the  partial  sums 
of  terms  containing  d'x  =  -  3,  -  2,  etc.  The  sum  of  the  numbers  is  26, 
which  agrees  with  the  result  obtained  above. 

The  rest  of  the  calculation  consists  of  substitution  in  known  formulas, 

and  is  given  below. 

l^fxd'x      -26  i:fyd'y      25    ,^,  __„, 

Cx  =  —^  =  -gQ-  Cy  =  ^^  =  —    (Theorem,  page  389) 

=  —  0.43  class  intervals.  =  0.42  class  intervals. 

(r,2  =  5^-  _  c,2  (7„2  =  ?^  -  c„2  (Theorem,  page  401) 

=  1^_(_0.43)^  =i-(0.42)^ 

=  1.82.  =  0.82. 

.*.  (Tx  =  1.3  class  intervals.  .*.  (Ty  =  0.91  class  intervals. 

By  (5),  Sd«dy  =  Sd'x  d'y  -  nCxCy 

=  26  -  60(-  0.43)  (0.42)  =  36.8. 

Then  by  (3),  ,  =  ^»  =  __3^^  =  0.52. 


428 


ELEMENTARY  FUNCTIONS 


By  (6)  the  probable  error  in  r  is 
P'Er  =  0.Q7 


1  -  0.52^      0.67  (1  +  0.52)  (1  -  0.52) 


\/60 
0.67x1.5x0.48 


V60 


Hence 
Since 
and 

we  have 
and 


11  =  0.062. 

V60 

r  =  0.52  ±  0.062. 

Cx  =  -  0.43  class  intervals  =  -  0.43  inches, 

Cy  =  0.42  class  intervals  =  2.1  bushels  per  acre, 

M^  =  E^  +  C:c  =  4.5  -  0.43  =  4.07  inches, 

My  =  Ey  +  Cy  =  32.5  +  2. 1  =  34.6  bushels  per  acre. 


The  slopes  of  the  hnes  of  regression  are,  by  (4), 


mi 


,^2  =  0.52  5^  =  0.36, 

(Ti  1.6 


and 


(Tl      ^.^   1.3 


0.74. 


The  entire  computation  has  been  conducted  in  terms  of  the  class  in- 
tervals as  units.  The  class  interval  on  the  x-axis  is  one  unit  and  that  on 
the  y-SLids  is  5  units.  Hence  the  ratio  of  the  class  intervals  is  5.  Then 
the  slopes  of  the  lines  in  the  units  used  in  the  figure  of  Example  1  are 

mi  =  0.36  X  5  =  1.8        and        m^  =-  ^  =  0.15. 
The  equations  of  the  lines  of  regression,  referred  to  the  point  (4.07, 
34.6)  as  origin  are  therefore 

y'  =  1.8a:'        and        x'  =  0.l5y'. 

EXERCISES 

1.  Calculate  the  coefficient  of  correlation  between  the  following  grades 
in  English  and  jilgebra,  and  determine  the  probable  error.  Find  the 
equations  of  the  lines  of  regression  and  plot  the  lines. 


Lower  ends  of 

.      English 

intervals 

55 

60 

65 

70 

75 

80 

85 

90 

95 

i 

95 
90 
85 
80 
75 
70 
65 
60 
55 

1 

1 
1 
1 

1 

1 
1 
3 
8 
3 
1 
2 
1 

2 

4 
7 
4 
5 
3 

1 

2 
5 

1 
7 
7 
3 
1 
3 
1 

1 
5 
10 
4 
3 
1 
3 

1 

2 
5 
1 
2 

2 

' 

THEORY  OF  MEASUREMENT 


429 


2.  The  following  scores  were  made  by  the  pupils  of  a  fourth  grade  in  a 
silent  reading  test.  The  first  number  in  each  set  is  the  number  of  words 
read  per  minute,  the  second  number  is  the  number  of  questions  answered 
in  five  minutes,  the  third  number  is  the  index  of  comprehension. 
(170,  39,  74),  (160,  29,  79),  (281,  31,  80),  (157,  35,  87),  (146,  27,  87),  (81, 
10,  89),  (345,  47,  90),  (160,  34,  90),  (88,  26,  91),  (233,  41,  92),  (191,  35,  93), 
(157,  32,  93),  (239,  49,  93),  (142,  42,  94),  (199,  38,  94),  (254,  51,  95), 
(194,  44,  95),  (208,  47,  95),  (155,  33,  96),  (206,  33,  96),  (508,  45,  97),  (208, 
41,  97),  (180,  45,  97),  (213,  45,  97),  (259,  51,  98),  (281,  52,  98),  (224,  34, 
100),  (186,  36,  100),  (141,  31,  100),  (153,  29,  100),  (147,  21,  100). 

Let  the  class  intervals  for  the  three  sets  be  8-12,  13-17,  etc.,  70-89, 
90-109,  etc.,  74-76,  77-79,  etc.,  respectively. 

Construct  a  correlation  table  with  the  words  read  as  x,  and  the  number 
of  questions  answered  as  y. 

Construct  a  correlation  table  letting  x  represent  the  number  of  questions 
answered  in  five  minutes  and  y  the  index  of  comprehension. 

Calculate  for  each  table  the  coefficient  of  correlation  and  its  probable 
error.    What  can  be  inferred  from  the  results? 

3.  The  following  table  gives  the  correlation  between  weight  of  seeds 
and  number  of  seeds  per  apple,  for  normal  apples. 

Determine  whether  a  sufiiciently  high  degree  of  correlation  exists  between 
the  number  and  weights  of  seeds  so  that  the  latter  may  be  used  instead  of 
the  former  in  the  study  of  the  relation  between  seed  and  pulp  development. 


H  (t  • 

1         1 

Seed  weight  in  milligrams 

Mm- values 

275 

325 

375 

425 

475 

525 

575 

625 

675 

725 

775 

825 

875 

1 

925 

975 

16 

1 

. . . 

15 

2 

4 

1 

1 

1 





2 
2 
3 

1 

2 
2 
1 

1 

14 

— 

13 

— 



2 
2 
5 

4 
4 

12 

— 

1 
1 
5 

7 

1 

7 
5 
3 

m 

11 

— 

2 

1 
3 
3 

O 

10 

flj 

— 

1 



^ 

9 

g 

— 

— 

3 

8 

^ 

T 

4 

4 

2 

7 



6 

2 

5 

4 

1 

430 


ELEMENTARY  FUNCTIONS 


4.  The  following  pairs  of  numbers  are  the  scores  made  by  pupils  of  a 
fourth  grade  in  reading  and  spelling  respectively. 

(20.7,  65),  (16.3,  65),  (15.1,  100),  (14.2,  85),  (14.2,  100),  (13,  95),  (12.5, 
70).  (12.2,  85),  (11.2,  75),  (10.5,  100),  (8.6,  55),  (7.8,  35),  (7.7.  40),  (7.5, 
60),  (7.4,  60),  (7.4,  75),  (7.2,  95),  (6.8,  95),  (6.6,  60),  (5.6,  40),  (6.4,  60), 
(5.4, 85),  (5.4, 45),  (5.2, 50),  (5.2, 75),  (5.2,  75),  (5.2, 95),  (4.7,  85),  (4.4.  65), 
(4.3,  45),  (4.2,  80),  (3.8,  55). 

Find  the  coeiEcient  of  correlation  and  its  probable  error. 

6.  Solve  Exercise  3  for  the  following  table. 


■\/fZ 

J 1 

Seed  weight  in  milligrams 

300 

340 

380 

420 

460 

500 

540 

580 

620 

660 

700 

740 

10 

111. 

1 

... 

... 

1 

1 

1 
4 

2 

1 

1 

2 

5 

1 

... 

-S 

9 

1 

4 

1 

2 

8 

... 

1 

1 

... 

2 
4 

7 

3 

3 

6 

2 

O 

7 

7 

6 

1 

2 

1 

6 

1 

1 

3 

1 

2 
2 

5 

1 

1 

1 

. . . 

. . . 

^ 

6 

. . . 

... 

... 

4 

3 

4 



,---,--,.--,...,-.- 

I'"' 

6.   Correlation  between  weight  of  seeds  and  weight  of  fruit,  for  normal 
apples.    Find  to  what  extent  seed  weight  is  related  to  weight  of  fruit. 


Fruit  weight  in  grams 

Mia-vaiues 

300 

340 

380 

420 

460 

500 

1 
1 
1 

540 

580 

620 

660 

700 

740 

195 

1 

... 

. . . 

. . . 

. . . 

185 

1 

1 

:& 

175 

. . . 

. . . 

1 

. . . 

1 

165 

1 

■  *  * 

1 

1 
1 

2 
1 

~4~ 
2 

1 

2 

1 

1 

a 

155 

... 

1 

1 

1 

.a 

.4^ 

145 

. . . 

1 
1 

^  1 

1 

1 
2 

1 

3 
3 
2 

1 

2 
1 
2 

1 

1 
1 

1 
1 

1 

_1_ 

1 

2 

§ 

135 

1 

1 

5 

1 

% 

125 

1 

3 

1 

1 

116 

1 

1 

3 

5 

1 

OQ 

105 

1 

2 

1 

4 

4 
2 

1 

— - 

•  •  • 

95 

... 

THEORY   OF  MEASUREMENT 


431 


7.  Distribution  of  100  pupils  in  each  grade  in  a  multiplication  test. 


Score 

0 

1 

2 

3 

4 

5 

6 

7 

8 

8-A 

1 

4 
5 

11 

20 
17 

23 

14 

17 

6 

4 

8-B 

1 

7 

24 

21 

16 

7 

2 

1 

7-A 

1 

6 

10 

18 

22 

19 

15 

8 

1 

7-B 

4 

10 

17 

20 

25 

10 

11 

3 

6-A 

3 

12 

17 

24 

22 

14 

7 

1 

6-B 

4 

13 

19 

26 

19 

12 

4 

3 

5-A 

12 

16 

28 

25 

12 

4 

2 

1 

5-B 

26 

20 

26 

21 

6 

1 

•• 

•• 

Determine  the  correlation  coefficient  and  its  probable  error.     Is  in- 
crease in  ability  in  multiplication  closely  related  to  progress  through  the 


8.  The  general  average  in  all  first  year  high  school  subjects  of  121 

students  and  the  records  made  by  them  in  English  and  algebra,  two  re- 
quired subjects,  are  given  in  the  table. 

Lower  ends  of  intervals           55,   60,   65,  70,     75,    80,     85,    90,  95 

General  average,  frequencies,    0      5      12  28      25      26      17      8  0 

English,                 frequencies,    1      4      20  26      30      28      10      2  0 

Algebra,                frequencies,    4      6      10  13      23      24      18    18  5 


If  one  subject  is  taken  as  an  index  of  the  general  ability  of  a  student 
is  it  "better  to  take  the  English  or  the  algebra  record? 

9.  Correlation  of  the  average  height  of  a  plant  of  oats  with  the  total 
yield  of  the  plant  in  grams. 


432 


ELEMENTARY  FUNCTIONS 


Yield  in  Grams 

0-1 

1-2 

2-3 

3-4 

4-5 

5-6 

6-7 

7-8 

8-9 

90-100 

... 

... 

5 

1 

... 

... 

85-90 

3 

7 

4 

3 

2 

5 

80-85 

4 

13 

20 

22 

27 

2 

— 

.§ 

75-80 

13 

23 

43 

32 

10 

2 

— 

70-75 

7 

46 

28 

15 

1 

0 

65-70 

13 

9 

9 

3 

60-65 

2 

1 

6 

7 

5 

1 

55-60 

6 

2 

- 

Ui 

50-55 

1 

1 

45-50 

... 

... 

Find  the  coefficient  of  correlation  and  plot  the  lines  of  regression. 
10.   Correlation  of  the  yield  of  a  plant  of  oats  with  the  number  of  kernels 
per  plant. 


Mid-values 

Kernels  per  plant 

25 

75 

125 

175 

225 

275 

325 

375 

425 

475 

8.5 

.. 

... 

... 

... 

1 

7.5 

1 

1 

a 

6.5 

.  • 

. . . 

3 

4 

. . . 

5.5 

12 

26 

4 

4.5 

4 

30 

39 

7 

? 

3.5 

4 

47 

51 

7 

., 

■^ 

2.5 

61 

45 

1.5 

20 

30 

.5 

2 

1 

... 

... 

... 

... 

... 

... 

Find  the  coefficient  of  correlation,  its  probable  error,  and  plot  the 
lines  of  regression.    What  plants  should  be  used  for  seeding  purposes? 


INDEX 


THE   NUMBERS   REFER  TO   PAGES 


Abridged  multiplication  and  divi- 
sion, 77. 
Abscissa,  14. 
Acceleration,    63,    276,    308,    312; 

composition    and    resolution    of, 

186. 
Algebraic  function,  38,  87. 
American  Experience  Table,  380. 
Amplitude    of   harmonic    function, 

359. 
Angle  of  elevation  or  depression, 

183. 
Annuities,  244. 
Approximate  error,  294,  414. 
Approximate   value   of  j{x  +  Ax), 

296. 
Arc  of  circle,  171. 
Area  under  a  curve,   305;    under 

probability  curve,  409,  412. 
Arithmetic  mean,  388. 
Astronomical  exercises,  235. 
Asymmetrical  distribution,  385. 
Asymptotes,  24,  27. 
Average,   83,   388;    ordinate,   307; 

rate  of  change,  35. 
Axes  of  coordinates,  15. 
Axis  of  symmetry,  22;  of  parabola, 

88. 

Bearing  of  line,  183. 
Binomial  expansion,  370. 
Biquadratic  function,  39. 

Center  of  symmetry,  22. 
Changes  of  function,  28,  272. 


Characteristic  of  logarithm,  227* 
properties  of  function,  42. 

Circle,  equation  of,  327. 

ClassiJfication  of  functions,  38. 

Coefficient  of  correlation,  421,  425; 
of  friction,  81,  189. 

Cofunction,  177. 

Combinations,  367,  371. 

Compound  harmonic  curve,  360;  in- 
terest, 241 ;  interest  law,  215. 

Components  of  acceleration,  187, 
312;  of  forces,  187;  of  velocities, 
187,  311. 

Composition  of  accelerations,  forces, 
and  velocities,  186. 

Cone,  325. 

Concavity,  273. 

Constant,  6;  of  integration,  302. 

Construction  of  tables,  297. 

Continuous  function,  269. 

Coordinates,  14. 

Correlation,  420. 

Cubic  function,  39. 

Curve  of  errors,  408. 

Cylinder,  318. 

Density,  61. 

Dependent  events,  375;  variable,  5. 

Derivative,  267,  274;  rules  for,  269, 

270,  271,  278,  280,  353,  354. 
Difference  of  sines  of  two  angles, 

345. 
Directed  line,  12. 
Discussion  of  table,  24. 
Division,  abridged,77;  synthetic,136. 


433 


434 


INDEX 


Empirical  data  problems,  by  least 
squares,  418;  exponential  func- 
tion, 257;  harmonic  fimction, 
361;  linear  function,  78;  power 
function,  127,  258;  quadratic 
function,  104. 

Equation  of  circle,  327;  of  straight 
line,  66;  of  probability  curve,  407. 

Equilibrium  of  a  particle,  188. 

Excluded  values,  22,  26. 

Exponential  equations,  237;  func- 
tion, 215;  graph  of,  216,  248,  255. 

Forces,  composition  and  resolution 
of,  186. 

Frequency  curve,  384,  385;  dis- 
tribution, 382;    polygon,  382. 

Function,  average  rate  of  change  of, 
35;  becoming  infinite,  25;  changes 
of,  28,  272;  classification  of,  38; 
defined  by  an  equation,  6;  defini- 
tion, 5;  derivative  of,  267;  dis- 
cussion of  table,  24;  fundamental 
problems  of,  41;  graph  of,  17; 
inverse  of,  40;  maximum  and 
minimmn  values,  29,  272;  nota- 
tion for,  8;  rate  of  change  of,  94, 
267,  272;  zeros  of,  23. 

Fundamental  trigonometric  formu- 
las, 332. 

Geometric  mean,  395. 

Graph  of  equation,  18;  exponential 
function,  216,  248,  255;  function, 
17;  harmonic  function,  358; 
linear  function,  57;  ploynomial, 
136;  power  function,  119,  254; 
quadratic  function,  100;  trig- 
onometric function,  172. 

Graph,  interpretation  of,  42. 

Graphs  of  inverse  functions,  114; 
reciprocal  functions,  118;  related 
functions,  151. 


Harmonic  curve,  358;    mean,  396; 

motion,  362. 
Histogram,  383. 
Horizontal  angle,  183. 
Horner's  method,  147. 
Hyperbola,  131. 

Identity,  134. 

Identities,  trigonometric,  336,  351. 
Inchnation  of  straight  line,  199. 
Independent  events,  375 ;  variable,  5. 
Infinite,  function  becoming,  25. 
Inflection,  point  of,  139,  273. 
Instantaneous  velocity,  93. 
Integral,  301;  rational  function,  39, 

133. 
Integration,  301;  rules  for,  303,  354. 
Intercepts,  23. 
Interpolation,  121,  178. 
Interpretation  of  graph,  42. 
Inverse  functions,   40;    graphs  of, 

114;  trigonometric  functions,  209. 
Irrational   function,   39;    roots   of 

equation,  147. 

Law  of  sines,  201;  of  cosines,  202. 
Least  squares,  417. 
Limit  of  a  variable,  93,  265.    --^ 
Linear    equation,    57;     fractional 

function,  131;   function,  39,  57. 
Lines  of  regression,  423. 
Logarithmic    function,    215,    221; 

graph  of,  222;  paper,  252;  scale, 

250;    solution  of  triangles,  233, 

346,  348. 
Logarithms,  common,  225. 

Mantissa,  227. 

Mathematical  expectation,  374. 
Maximum  point,  29,  272. 
Maxima  and  minima,  285. 
Mean,  arithmetic,  388;    deviation, 

399;    geometric,  395;    harmonic, 

396. 


INDEX 


435 


Median,  391;   deviation,  400 
Minimum  point,  29,  272. 
Mode,  392. 
MortaUty  table,  380. 
Mutually  exclusive  events,  376. 

Nonnal  line,  281. 

Oblique  triangles,  203,233,  346,  348. 
Octahedron,  regular,  330. 
Ordinate,  15. 
Origin,  15. 

Parallelogram  law,  186. 

Parallelopiped,  316. 

Parabola,  88,  100. 

Period  of  a  function,  168;  of  har- 
monic function,  359;  of  sin  0, 174; 
of  sin  2$,  342;  of  tan  d,  195. 

Periodic  function,  168. 

Permutations,   366. 

Point  of  inflection,  139,  273. 

Point-slope  equation  of  straight 
line,  67. 

Polyhedron,  316. 

Polynomial,  39,  133 

Power  function,  107;  graph  of,  119, 
254;  properties  of,  153. 

Present  value,  242;  of  annuity,  245. 

Principal  value  of  inverse  trig- 
onometric function,  210. 

Prism,  315,  317. 

Probable  error,  400,  411;  of  co- 
efficient of  correlation,   '26. 

ProbabiUty,  373;  curve,  407. 

Proportional  variables,  61,  125. 

Quadratic  function,  39,  100. 
Quartile  deviation,  399. 

Radian,  171. 

Rate  of  change,  94;  average,  35; 
of  polynomial,  138,  271;  uni- 
form, 48. 


Rational  function,  39. 

Rational  roots  of  equations,  140. 

Reciprocal  function,  118;  relations 
of  trigonometric  functions,  167. 

Rectangular  parallelopiped,  316. 

Regular  tetrahedron,  329;  octa- 
hedron, 330. 

Related  functions,  151;  rates,  289. 

Relative  error,  73,  291. 

Remainder  theorem,  134. 

Resolution  of  accelerations,  forces, 
velocities,  187. 

Revolution,  solid  of,  323. 

Right  triangles,  180,  233. 

Semi-logarithmic  paper,  252. 

Sign  of  function,  28. 

Significant  figures,  73. 

Simple  harmonic  motion,  362. 

Slide  rule,  251. 

Slope  of  straight  line,  50,  51,  199; 
of  parallel  lines,  52;  of  perpen- 
dicular lines,  200;  of  tangent 
line,  95,  281. 

Slope-intercept  equation  of  straight 
line,  66. 

Small  errors,  291. 

Snow-ball  law,  215. 

Solid  of  revolution,  323. 

Solution  of  triangles,  181,  203,  233, 
346,  348. 

Sphere,  327. 

Standard  deviation,  400;  form  of 
numbers,  73. 

Sum  of  sines  of  two  angles,  345. 

Sum  of  two  angles,  164;  two  lines, 
13. 

Symmetrical  distribution,  385,  404. 

Symmetry  with  respect  to  point  or 
Une,  15,  22-24. 

Synthetic  division,  ,136. 

Tables,  construction  of,  297. 
Tabular  difference,  121. 


436 


INDEX 


Tangent  line,  95,  281. 
Tetrahedron,  regular,  329. 
Translation  of  axes,  89;   of  y-axis, 

144. 
Transcendental  functions,  39. 
Trigonometric  equations,  334,  351. 
Trigonometric  functions,  definitions, 

166;   of^+</),  338;   of  ^-<^,  341; 

of    2d,    342;     of    0/2,    343;     of 

n90°  ±  e,  177,  192. 
Trigonometric  identities,  336,  351. 


in  a  circle,  354;   rate  of  change, 
48;   scale,  250;    velocity,  46. 

Variable,   5;    dependent,   5;    inde- 
pendent, 5. 
Variation,  61,  125;  of  function,  31. 

Velocity  at  an  instant,  93,  276,  308, 
312;  composition  and  resolution 
of,  186;    uniform,  46. 


Weighted  arithmetic  mean,  389. 
Uniform  acceleration,  63;    motion      Zeros  of  function,  23. 


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